Tuesday, December 12, 2017

Number of the day: 732499

Properties of the number 732499:

732499 is a cyclic number.
732499 = 31 × 23629 is semiprime and squarefree.
732499 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 708840 totatives.
732499 has a semiprime digit sum 34 in base 10.
732499 has a Fibonacci digit sum 34 in base 10.
Reversing the decimal digits of 732499 results in a prime.
732499 = 3662502 - 3662492 = 118302 - 117992 is the difference of 2 nonnegative squares in 2 ways.
732499 is the difference of 2 positive pentagonal numbers in 2 ways.
732499 = 332 + 1032 + 8492 is the sum of 3 positive squares.
7324992 = 874512 + 7272602 is the sum of 2 positive squares in 1 way.
7324992 is the sum of 3 positive squares.
732499 is a proper divisor of 1373660 - 1.
732499 = '73249' + '9' is the concatenation of 2 semiprime numbers.
732499 is an emirpimes in (at least) the following bases: 3, 4, 7, 8, 13, 16, 17, 22, 28, 29, 33, 35, 38, 39, 41, 49, 50, 58, 60, 62, 63, 66, 67, 68, 69, 73, 74, 77, 79, 80, 82, 83, 89, 90, 91, 96, and 100.
732499 is palindromic in (at least) the following bases: 26, and -92.
732499 in base 5 = 141414444 and consists of only the digits '1' and '4'.

Monday, December 11, 2017

Number of the day: 892849

Properties of the number 892849:

892849 is a cyclic number.
892849 is the 70763th prime.
892849 has 13 antidivisors and 892848 totatives.
Reversing the decimal digits of 892849 results in a sphenic number.
892849 = 4464252 - 4464242 is the difference of 2 nonnegative squares in 1 way.
892849 is the sum of 2 positive triangular numbers.
892849 is the difference of 2 positive pentagonal numbers in 1 way.
892849 = 602 + 9432 is the sum of 2 positive squares in 1 way.
892849 = 2182 + 3872 + 8342 is the sum of 3 positive squares.
8928492 = 1131602 + 8856492 is the sum of 2 positive squares in 1 way.
8928492 is the sum of 3 positive squares.
892849 is a proper divisor of 1741979 - 1.
892849 is an emirp in (at least) the following bases: 5, 13, 14, 15, 22, 25, 28, 29, 40, 59, 75, 81, 86, and 99.

Sunday, December 10, 2017

Number of the day: 77679

Carl Gustav Jacob Jacobi was born on this day 213 years ago.

Properties of the number 77679:

77679 = 34 × 7 × 137 is the 70046th composite number and is not squarefree.
77679 has 3 distinct prime factors, 20 divisors, 23 antidivisors and 44064 totatives.
77679 has a triangular digit sum 36 in base 10.
77679 = 63 + 153 + 423 is the sum of 3 positive cubes in 1 way.
77679 is the difference of 2 nonnegative squares in 10 ways.
77679 is the difference of 2 positive pentagonal numbers in 1 way.
77679 is not the sum of 3 positive squares.
776792 = 498962 + 595352 is the sum of 2 positive squares in 1 way.
776792 is the sum of 3 positive squares.
77679 is a proper divisor of 109718 - 1.
77679 = '77' + '679' is the concatenation of 2 semiprime numbers.
77679 is palindromic in (at least) the following bases: 47, 90, and -54.
77679 in base 47 = Z7Z and consists of only the digits '7' and 'Z'.
77679 in base 53 = RYY and consists of only the digits 'R' and 'Y'.

The number 77679 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A211545: Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>0.
A274361: Numbers n such that n and n+1 both have 20 divisors.
A293440: First differences of A293230: how many more alive nodes there are in generation n+1 than in generation n in the binary tree of persistently squarefree numbers.

Saturday, December 9, 2017

Number of the day: 4815

Properties of the number 4815:

4815 = 32 × 5 × 107 is the 4166th composite number and is not squarefree.
4815 has 3 distinct prime factors, 12 divisors, 11 antidivisors and 2544 totatives.
4815 = 292 + … + 332 is the sum of at least 2 consecutive positive squares in 1 way.
4815 = 73 + 123 + 143 is the sum of 3 positive cubes in 1 way.
4815 = 24082 - 24072 = 8042 - 8012 = 4842 - 4792 = 2722 - 2632 = 1682 - 1532 = 762 - 312 is the difference of 2 nonnegative squares in 6 ways.
4815 is the difference of 2 positive pentagonal numbers in 1 way.
4815 is not the sum of 3 positive squares.
48152 = 28892 + 38522 is the sum of 2 positive squares in 1 way.
48152 is the sum of 3 positive squares.
4815 is a proper divisor of 6416 - 1.
4815 = '4' + '815' is the concatenation of 2 semiprime numbers.
4815 is palindromic in (at least) the following bases: 6, 58, and -83.
4815 in base 24 = 88f and consists of only the digits '8' and 'f'.
4815 in base 57 = 1RR and consists of only the digits '1' and 'R'.
4815 in base 58 = 1P1 and consists of only the digits '1' and 'P'.

The number 4815 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001522: Number of n-stacks with strictly receding walls, or planar partitions of n.
A027578: Sums of five consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2.
A051865: 13-gonal (or tridecagonal) numbers: a(n) = n*(11*n - 9)/2.
A090669: Numbers n such that 7^n - 2 is a prime.
A121035: Multiples of 15 containing a 15 in their decimal representation.
A131246: Row sums of triangle A131245.
A226514: Column 3 of array in A226513.
A238128: Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest descent k, n>=0, 0<=k<=n.
A257211: Numbers n such that the decimal expansions of both n and n^2 have 1 as smallest digit and 8 as largest digit.
A275172: Positions of 3 in A274640.

Friday, December 8, 2017

Number of the day: 429203302

Jacques Salomon Hadamard was born on this day 152 years ago.

Julia Robinson was born on this day 98 years ago.

Properties of the number 429203302:

429203302 = 2 × 11 × 19509241 is a sphenic number and squarefree.
429203302 has 3 distinct prime factors, 8 divisors, 39 antidivisors and 195092400 totatives.
429203302 has a semiprime digit sum 25 in base 10.
429203302 is the sum of 2 positive triangular numbers.
429203302 is the difference of 2 positive pentagonal numbers in 2 ways.
429203302 = 862 + 2912 + 207152 is the sum of 3 positive squares.
4292033022 = 3014441982 + 3055272002 is the sum of 2 positive squares in 1 way.
4292033022 is the sum of 3 positive squares.
429203302 is a proper divisor of 683162577 - 1.
429203302 = '429203' + '302' is the concatenation of 2 semiprime numbers.
429203302 = '42' + '9203302' is the concatenation of 2 sphenic numbers.

Thursday, December 7, 2017

Number of the day: 696787

Properties of the number 696787:

696787 = 7 × 132 × 19 × 31 is the 640473th composite number and is not squarefree.
696787 has 4 distinct prime factors, 24 divisors, 47 antidivisors and 505440 totatives.
696787 has a prime digit sum 43 in base 10.
696787 = 833 + 503 is the sum of 2 positive cubes in 1 way.
696787 is the difference of 2 nonnegative squares in 12 ways.
696787 is the sum of 2 positive triangular numbers.
696787 is the difference of 2 positive pentagonal numbers in 12 ways.
696787 = 392 + 1212 + 8252 is the sum of 3 positive squares.
6967872 = 2679952 + 6431882 = 4906372 + 4947602 is the sum of 2 positive squares in 2 ways.
6967872 is the sum of 3 positive squares.
696787 is a proper divisor of 1913 - 1.
696787 = '69' + '6787' is the concatenation of 2 semiprime numbers.

The number 696787 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001872: Convolved Fibonacci numbers.
A061179: Fourth column (m=3) of triangle A060920 (bisection of Fibonacci triangle, even part).
A163763: Sqrt(sigma(A008847(n)^2)), where A008847 lists m such that sigma(m^2) is a square.

Wednesday, December 6, 2017

Number of the day: 1666

Properties of the number 1666:

1666 = 2 × 72 × 17 is the 1404th composite number and is not squarefree.
1666 has 3 distinct prime factors, 12 divisors, 11 antidivisors and 672 totatives.
1666 has a prime digit sum 19 in base 10.
Reversing the decimal digits of 1666 results in a prime.
1666 is the difference of 2 positive pentagonal numbers in 4 ways.
1666 = 212 + 352 is the sum of 2 positive squares in 1 way.
1666 = 12 + 242 + 332 is the sum of 3 positive squares.
16662 = 7842 + 14702 is the sum of 2 positive squares in 1 way.
16662 is the sum of 3 positive squares.
1666 is a proper divisor of 8832 - 1.
1666 = '166' + '6' is the concatenation of 2 semiprime numbers.
1666 = '1' + '666' is the concatenation of 2 triangular numbers.
1666 is palindromic in (at least) the following bases: 26, 37, 48, 97, -32, and -45.
1666 in base 6 = 11414 and consists of only the digits '1' and '4'.
1666 consists of only the digits '1' and '6'.
1666 in base 23 = 33a and consists of only the digits '3' and 'a'.
1666 in base 25 = 2gg and consists of only the digits '2' and 'g'.
1666 in base 26 = 2c2 and consists of only the digits '2' and 'c'.
1666 in base 36 = 1aa and consists of only the digits '1' and 'a'.
1666 in base 37 = 181 and consists of only the digits '1' and '8'.
1666 in base 40 = 11Q and consists of only the digits '1' and 'Q'.

The number 1666 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A026798: Number of partitions of n in which the least part is 5.
A033547: Otto Haxel's guess for magic numbers of nuclear shells.
A035607: Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d>=1, m >= 0).
A054252: Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.
A132760: a(n) = n(n+15).
A145540: Number of numbers removed in each step of Eratosthenes' sieve for 10^4.
A185325: Number of partitions of n into parts >= 5.
A191726: Dispersion of A047216, (numbers >1 and congruent to 1 or 2 mod 5), by antidiagonals.
A195313: Generalized 13-gonal numbers: n*(11*n-9)/2, n=0,1,-1,2,-2,...
A292456: Numbers where 6 outnumbers any other digit.

Monday, December 4, 2017

Number of the day: 629435

Properties of the number 629435:

629435 is a cyclic number.
629435 = 5 × 125887 is semiprime and squarefree.
629435 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 503544 totatives.
629435 has a prime digit sum 29 in base 10.
629435 has an oblong digit product 6480 in base 10.
629435 = 3147182 - 3147172 = 629462 - 629412 is the difference of 2 nonnegative squares in 2 ways.
629435 is the difference of 2 positive pentagonal numbers in 2 ways.
629435 = 152 + 192 + 7932 is the sum of 3 positive squares.
6294352 = 3776612 + 5035482 is the sum of 2 positive squares in 1 way.
6294352 is the sum of 3 positive squares.
629435 is a proper divisor of 15120981 - 1.
629435 is an emirpimes in (at least) the following bases: 3, 7, 9, 11, 12, 15, 16, 19, 23, 25, 26, 31, 33, 35, 38, 44, 47, 48, 50, 53, 55, 61, 63, 65, 69, 70, 72, 73, 75, 76, 77, 82, 84, 92, 94, 95, 97, and 98.

Sunday, December 3, 2017

Number of the day: 64118

Properties of the number 64118:

64118 = 2 × 32059 is semiprime and squarefree.
64118 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 32058 totatives.
64118 has an oblong digit sum 20 in base 10.
Reversing the decimal digits of 64118 results in a sphenic number.
64118 = 32 + 102 + 2532 is the sum of 3 positive squares.
641182 is the sum of 3 positive squares.
64118 is a proper divisor of 190139 - 1.
64118 is an emirpimes in (at least) the following bases: 2, 4, 5, 6, 11, 19, 22, 23, 26, 29, 31, 36, 44, 47, 52, 55, 63, 67, 68, 69, 71, 75, 78, 79, 82, 87, 90, 92, 94, and 97.
64118 is palindromic in (at least) base 49.
64118 in base 49 = QYQ and consists of only the digits 'Q' and 'Y'.

Saturday, December 2, 2017

Number of the day: 887818

Properties of the number 887818:

887818 = 2 × 443909 is semiprime and squarefree.
887818 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 443908 totatives.
887818 is the difference of 2 positive pentagonal numbers in 2 ways.
887818 = 5332 + 7772 is the sum of 2 positive squares in 1 way.
887818 = 882 + 2072 + 9152 is the sum of 3 positive squares.
8878182 = 3196402 + 8282822 is the sum of 2 positive squares in 1 way.
8878182 is the sum of 3 positive squares.
887818 is a proper divisor of 7110977 - 1.
887818 is an emirpimes in (at least) the following bases: 8, 13, 14, 15, 17, 19, 21, 25, 27, 32, 34, 40, 41, 46, 50, 52, 54, 56, 58, 61, 62, 75, 81, 84, 86, 88, 92, 93, and 97.

Friday, December 1, 2017

Number of the day: 285694

Nikolai Lobachevsky was born on this day 225 years ago.

Properties of the number 285694:

285694 = 2 × 211 × 677 is a sphenic number and squarefree.
285694 has 3 distinct prime factors, 8 divisors, 51 antidivisors and 141960 totatives.
285694 has a semiprime digit sum 34 in base 10.
285694 has a Fibonacci digit sum 34 in base 10.
Reversing the decimal digits of 285694 results in a semiprime.
285694 is the sum of 2 positive triangular numbers.
285694 is the difference of 2 positive pentagonal numbers in 3 ways.
285694 = 222 + 572 + 5312 is the sum of 3 positive squares.
2856942 = 219442 + 2848502 is the sum of 2 positive squares in 1 way.
2856942 is the sum of 3 positive squares.
285694 is a proper divisor of 181191 - 1.
285694 is palindromic in (at least) the following bases: -7, and -92.
285694 in base 26 = g6g6 and consists of only the digits '6' and 'g'.

The number 285694 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A173674: ceil[A003269(n)/2].

Thursday, November 30, 2017

Number of the day: 330395165736

Properties of the number 330395165736:

330395165736 = 23 × 3 × 181 × 293 × 259583 is composite and not squarefree.
330395165736 has 5 distinct prime factors, 64 divisors, 43 antidivisors and 109149039360 totatives.
330395165736 has an emirpimes digit sum 51 in base 10.
Reversing the decimal digits of 330395165736 results in a sphenic number.
330395165736 is the difference of 2 nonnegative squares in 16 ways.
330395165736 is the sum of 2 positive triangular numbers.
330395165736 is the difference of 2 positive pentagonal numbers in 3 ways.
330395165736 = 1942 + 66162 + 5747622 is the sum of 3 positive squares.
3303951657362 = 1099905087602 + 3115494399362 = 346823654642 + 3285697780802 = 425196954002 + 3276477392642 = 766787415362 + 3213741373202 is the sum of 2 positive squares in 4 ways.
3303951657362 is the sum of 3 positive squares.
330395165736 is a proper divisor of 17091593882 - 1.

Tuesday, November 28, 2017

Number of the day: 65596

Properties of the number 65596:

65596 = 22 × 232 × 31 is the 59044th composite number and is not squarefree.
65596 has 3 distinct prime factors, 18 divisors, 21 antidivisors and 30360 totatives.
65596 has an emirp digit sum 31 in base 10.
65596 = 164002 - 163982 = 7362 - 6902 = 5602 - 4982 is the difference of 2 nonnegative squares in 3 ways.
65596 is the sum of 2 positive triangular numbers.
65596 is the difference of 2 positive pentagonal numbers in 3 ways.
65596 is not the sum of 3 positive squares.
655962 is the sum of 3 positive squares.
65596 is a proper divisor of 55722 - 1.
65596 = '6559' + '6' is the concatenation of 2 semiprime numbers.
65596 is palindromic in (at least) the following bases: 48, 52, and 56.
65596 in base 46 = V00 and consists of only the digits '0' and 'V'.
65596 in base 48 = SMS and consists of only the digits 'M' and 'S'.
65596 in base 52 = ODO and consists of only the digits 'D' and 'O'.
65596 in base 56 = KpK and consists of only the digits 'K' and 'p'.

The number 65596 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A114359: Let M(n) be the n X n matrix m(i,j)=min(i,j) for 1<=i,j<=n then a(n) is the trace of M(n)^(-7).
A177269: a(n) = 1 + sum{k=1 to n} A177268(k)

Monday, November 27, 2017

Number of the day: 782975

Properties of the number 782975:

782975 = 52 × 31319 is the 720289th composite number and is not squarefree.
782975 has 2 distinct prime factors, 6 divisors, 21 antidivisors and 626360 totatives.
782975 has a semiprime digit sum 38 in base 10.
Reversing the decimal digits of 782975 results in a prime.
782975 = 3914882 - 3914872 = 783002 - 782952 = 156722 - 156472 is the difference of 2 nonnegative squares in 3 ways.
782975 is the sum of 2 positive triangular numbers.
782975 is the difference of 2 positive pentagonal numbers in 3 ways.
782975 is not the sum of 3 positive squares.
7829752 = 4697852 + 6263802 = 2192332 + 7516562 is the sum of 2 positive squares in 2 ways.
7829752 is the sum of 3 positive squares.
782975 is a proper divisor of 1012237 - 1.

Sunday, November 26, 2017

Number of the day: 36440

Norbert Wiener was born on this day 123 years ago.

Properties of the number 36440:

36440 = 23 × 5 × 911 is the 32577th composite number and is not squarefree.
36440 has 3 distinct prime factors, 16 divisors, 11 antidivisors and 14560 totatives.
36440 has an emirp digit sum 17 in base 10.
36440 = 402 + … + 552 is the sum of at least 2 consecutive positive squares in 1 way.
36440 = 91112 - 91092 = 45572 - 45532 = 18272 - 18172 = 9212 - 9012 is the difference of 2 nonnegative squares in 4 ways.
36440 is the difference of 2 positive pentagonal numbers in 2 ways.
36440 = 202 + 382 + 1862 is the sum of 3 positive squares.
364402 = 218642 + 291522 is the sum of 2 positive squares in 1 way.
364402 is the sum of 3 positive squares.
36440 is a proper divisor of 18234 - 1.
36440 is palindromic in (at least) the following bases: 43, 66, -69, and -72.
36440 in base 3 = 1211222122 and consists of only the digits '1' and '2'.
36440 in base 19 = 55hh and consists of only the digits '5' and 'h'.
36440 in base 43 = JUJ and consists of only the digits 'J' and 'U'.

Friday, November 24, 2017

Number of the day: 5822

Properties of the number 5822:

5822 = 2 × 41 × 71 is a sphenic number and squarefree.
5822 has 3 distinct prime factors, 8 divisors, 11 antidivisors and 2800 totatives.
5822 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 5822 results in a semiprime.
5822 is the difference of 2 positive pentagonal numbers in 1 way.
5822 = 12 + 142 + 752 is the sum of 3 positive squares.
58222 = 12782 + 56802 is the sum of 2 positive squares in 1 way.
58222 is the sum of 3 positive squares.
5822 is a proper divisor of 8575 - 1.
5822 = '58' + '22' is the concatenation of 2 semiprime numbers.
5822 is palindromic in (at least) the following bases: 81, and -60.
5822 in base 3 = 21222122 and consists of only the digits '1' and '2'.
5822 in base 9 = 7878 and consists of only the digits '7' and '8'.
5822 in base 18 = hh8 and consists of only the digits '8' and 'h'.

The number 5822 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000712: Number of partitions of n into parts of 2 kinds.
A048788: a(2n+1) = a(2n) + a(2n-1), a(2n) = 2*a(2n-1) + a(2n-2).
A052530: a(0)=0, a(1)=2; for n>=2, a(n)=4*a(n-1)-a(n-2).
A082630: Start with the sequence S(0)={1,1} and for n > 0 define S(n) to be I(S(n-1)) where I denotes the operation of inserting, for i=1,2,3..., the term a(i) + a(i+1) between any two terms for which 7*a(i+1) <= 11*a(i). The listed terms are the initial terms of the limit of this process.
A097097: Antidiagonal sums of triangle A097094, where self-convolution forms A097096 (row sums of triangle A097094).
A107235: Expansion of 1 / Prod{n>=0} (1-q^(5n+1))(1-q^(5n+2))(1-q^(5n+4)).
A152528: a(n) = p(n)*p(n+2) - 3*p(n+1), where p(n) is the n-th prime.
A208960: T(n,k)=Number of nXk 0..5 arrays with every element value z a city block distance of exactly z from another element value z
A218471: a(n) = n*(7*n-3)/2.
A221255: T(n,k)=Number of nXk arrays of occupancy after each element moves to some horizontal or antidiagonal neighbor, with no occupancy greater than 2

Thursday, November 23, 2017

Number of the day: 11232017

John Wallis was born on this day 401 years ago.

Happy Fibonacci Day!

Properties of the number 11232017:

11232017 is a cyclic number.
11232017 is the 740820th prime.
11232017 has 13 antidivisors and 11232016 totatives.
11232017 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 11232017 results in a sphenic number.
11232017 = 56160092 - 56160082 is the difference of 2 nonnegative squares in 1 way.
11232017 is the difference of 2 positive pentagonal numbers in 1 way.
11232017 = 14962 + 29992 is the sum of 2 positive squares in 1 way.
11232017 = 552 + 2162 + 33442 is the sum of 3 positive squares.
112320172 = 67559852 + 89730082 is the sum of 2 positive squares in 1 way.
112320172 is the sum of 3 positive squares.
11232017 is a proper divisor of 77337946 - 1.
11232017 = '1123' + '2017' is the concatenation of 2 prime numbers.
11232017 is an emirp in (at least) the following bases: 2, 7, 24, 31, 37, 38, 47, 50, 55, 62, 63, 75, 76, 79, 80, and 97.

Wednesday, November 22, 2017

Number of the day: 4371020300

Properties of the number 4371020300:

4371020300 = 22 × 52 × 19 × 401 × 5737 is the 4164311888th composite number and is not squarefree.
4371020300 has 5 distinct prime factors, 72 divisors, 35 antidivisors and 1651968000 totatives.
4371020300 has an oblong digit sum 20 in base 10.
4371020300 is the difference of 2 nonnegative squares in 12 ways.
4371020300 is the difference of 2 positive pentagonal numbers in 12 ways.
4371020300 = 5302 + 8582 + 661062 is the sum of 3 positive squares.
43710203002 = 22607222202 + 37409829602 = 19020079202 + 39355030602 = 25997335202 + 35138588602 = 4360120002 + 43492197002 = 285285002 + 43709272002 = 8396955002 + 42896072002 = 16363530362 + 40531675522 = 12512469762 + 41881021322 = 20071976962 + 38829081722 = 29583414202 + 32177685602 = 26453791202 + 34796246602 = 29278684602 + 32455207202 = 7992099962 + 42973342722 = 3949823362 + 43531376522 = 11964722562 + 42040780922 = 26226121802 + 34968162402 = 22850904802 + 37261481402 = 29372768802 + 32370083402 = 4076165002 + 43519728002 = 12238856842 + 41961794882 = 8272405442 + 42920265082 = 16098642242 + 40637612682 is the sum of 2 positive squares in 22 ways.
43710203002 is the sum of 3 positive squares.
4371020300 is a proper divisor of 4574780 - 1.

Monday, November 20, 2017

Number of the day: 10991

Benoit Mandelbrot was born on this day 93 years ago.

Properties of the number 10991:

10991 is a cyclic number.
10991 = 29 × 379 is semiprime and squarefree.
10991 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 10584 totatives.
10991 has an oblong digit sum 20 in base 10.
Reversing the decimal digits of 10991 results in an emirpimes.
10991 = 223 + 73 is the sum of 2 positive cubes in 1 way.
10991 = 54962 - 54952 = 2042 - 1752 is the difference of 2 nonnegative squares in 2 ways.
10991 is the sum of 2 positive triangular numbers.
10991 is the difference of 2 positive pentagonal numbers in 2 ways.
10991 is not the sum of 3 positive squares.
109912 = 75802 + 79592 is the sum of 2 positive squares in 1 way.
109912 is the sum of 3 positive squares.
10991 is a proper divisor of 15673 - 1.
10991 is an emirpimes in (at least) the following bases: 2, 4, 8, 10, 11, 15, 17, 18, 19, 20, 22, 25, 27, 31, 32, 41, 43, 46, 47, 49, 54, 61, 65, 67, 68, 74, 76, 77, 80, 81, 82, 84, 85, 87, 90, 91, 97, 99, and 100.
10991 is palindromic in (at least) the following bases: 23, 34, -9, -67, and -99.
10991 in base 4 = 2223233 and consists of only the digits '2' and '3'.
10991 in base 23 = khk and consists of only the digits 'h' and 'k'.
10991 in base 27 = f22 and consists of only the digits '2' and 'f'.
10991 in base 34 = 9h9 and consists of only the digits '9' and 'h'.
10991 in base 60 = 33B and consists of only the digits '3' and 'B'.

The number 10991 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A023192: Conjecturally, number of infinitely-recurring prime patterns on n consecutive integers.
A028948: An "extremely strange sequence": a(n+1) = [ A*a(n)+B ]/p^r, where p^r is the highest power of p diving [ A*a(n)+B ] and p=2, A=4.001, B=1.2.
A060354: The n-th n-gonal number.
A066831: Numbers n such that sigma(n) divides sigma(phi(n)).
A067383: Numbers n such that sigma(phi(n))/sigma(n) = 3.
A085366: Semiprimes that are the sum of two positive cubes. Common terms of A003325 and A046315.
A224483: Numbers which are the sum of two positive cubes and divisible by 29.
A252039: T(n,k)=Number of (n+2)X(k+2) 0..4 arrays with every 3X3 subblock row and column sum nonprime and every diagonal and antidiagonal sum prime
A255187: 29-gonal numbers: a(n) = n*(27*n-25)/2.
A262909: a(n) = greatest k such that A155043(k+A262509(n)) < A155043(A262509(n)).

Sunday, November 19, 2017

Number of the day: 15987776874

Properties of the number 15987776874:

15987776874 = 2 × 3 × 97 × 27470407 is composite and squarefree.
15987776874 has 4 distinct prime factors, 16 divisors, 13 antidivisors and 5274317952 totatives.
15987776874 has a semiprime digit sum 69 in base 10.
15987776874 = 11922 + 21432 + 1264192 is the sum of 3 positive squares.
159877768742 = 107134587302 + 118672158242 is the sum of 2 positive squares in 1 way.
159877768742 is the sum of 3 positive squares.
15987776874 is a proper divisor of 1939156802 - 1.
15987776874 = '1598' + '7776874' is the concatenation of 2 sphenic numbers.

Saturday, November 18, 2017

Number of the day: 879706944

Properties of the number 879706944:

879706944 = 26 × 32 × 23 × 66403 is the 834681825th composite number and is not squarefree.
879706944 has 4 distinct prime factors, 84 divisors, 31 antidivisors and 280482048 totatives.
879706944 is the difference of 2 nonnegative squares in 30 ways.
879706944 is the sum of 2 positive triangular numbers.
879706944 is the difference of 2 positive pentagonal numbers in 2 ways.
879706944 = 28002 + 48882 + 291202 is the sum of 3 positive squares.
8797069442 is the sum of 3 positive squares.
879706944 is a proper divisor of 2714092 - 1.

Thursday, November 16, 2017

Number of the day: 68956

Properties of the number 68956:

68956 = 22 × 17239 is the 62103th composite number and is not squarefree.
68956 has 2 distinct prime factors, 6 divisors, 3 antidivisors and 34476 totatives.
68956 has a semiprime digit sum 34 in base 10.
68956 has a Fibonacci digit sum 34 in base 10.
Reversing the decimal digits of 68956 results in a semiprime.
68956 = 23 + 33 + 413 is the sum of 3 positive cubes in 1 way.
68956 = 172402 - 172382 is the difference of 2 nonnegative squares in 1 way.
68956 is the difference of 2 positive pentagonal numbers in 1 way.
68956 is not the sum of 3 positive squares.
689562 is the sum of 3 positive squares.
68956 is a proper divisor of 50917 - 1.
68956 in base 51 = QQ4 and consists of only the digits '4' and 'Q'.

The number 68956 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A032375: Numbers n such that 51*2^n+1 is prime.
A184642: Number of partitions of n having no parts with multiplicity 7.

Wednesday, November 15, 2017

Number of the day: 5471

Properties of the number 5471:

5471 is a cyclic number.
5471 is the 722th prime.
5471 has 9 antidivisors and 5470 totatives.
5471 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 5471 results in a semiprime.
5471 = 27362 - 27352 is the difference of 2 nonnegative squares in 1 way.
5471 is the difference of 2 positive pentagonal numbers in 1 way.
5471 is not the sum of 3 positive squares.
54712 is the sum of 3 positive squares.
5471 is a proper divisor of 2547 - 1.
5471 is an emirp in (at least) the following bases: 3, 5, 7, 9, 11, 13, 14, 16, 19, 20, 32, 33, 41, 49, 51, 55, 56, 57, 59, 61, 62, 64, 65, 66, 68, 69, 74, 77, 81, 83, 85, 86, 87, 91, 93, and 98.
5471 is palindromic in (at least) base -18.
5471 in base 3 = 21111122 and consists of only the digits '1' and '2'.
5471 in base 4 = 1111133 and consists of only the digits '1' and '3'.
5471 in base 20 = ddb and consists of only the digits 'b' and 'd'.

The number 5471 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000945: Euclid-Mullin sequence: a(1) = 2, a(n+1) is smallest prime factor of 1 + Product_{k=1..n} a(k).
A049438: p, p+6 and p+8 are all primes (A046138) but p+2 is not.
A059802: Numbers n such that 5^n - 4^n is prime.
A078853: Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d = 2, 4 or 6) and forming d-pattern=[6,2,4]; short d-string notation of pattern = [624].
A124179: Prime(R(p)) where Prime(i) is the i-th prime and R(p) is the value of the reverse of the digits of prime p.
A136073: Father primes of order 4.
A167054: Values of A167053(k)-A167053(k-1)-1 not equal to 1.
A213978: Number of solid standard Young tableaux of shape [[n,n,n],[n]].
A214722: Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.
A250755: T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing x(i,j)+x(i-1,j) in the j direction

Tuesday, November 14, 2017

Number of the day: 8049637

Properties of the number 8049637:

8049637 is a cyclic number.
8049637 = 73 × 110269 is semiprime and squarefree.
8049637 has 2 distinct prime factors, 4 divisors, 13 antidivisors and 7939296 totatives.
8049637 has an emirp digit sum 37 in base 10.
8049637 = 40248192 - 40248182 = 551712 - 550982 is the difference of 2 nonnegative squares in 2 ways.
8049637 is the sum of 2 positive triangular numbers.
8049637 is the difference of 2 positive pentagonal numbers in 2 ways.
8049637 = 12862 + 25292 = 6942 + 27512 is the sum of 2 positive squares in 2 ways.
8049637 = 1002 + 2312 + 28262 is the sum of 3 positive squares.
80496372 = 52929122 + 60647952 = 47420452 + 65045882 = 38183882 + 70863652 = 17826602 + 78497632 is the sum of 2 positive squares in 4 ways.
80496372 is the sum of 3 positive squares.
8049637 is a proper divisor of 15973063 - 1.
8049637 is an emirpimes in (at least) the following bases: 3, 5, 12, 13, 14, 15, 16, 17, 20, 21, 29, 31, 38, 41, 44, 49, 51, 55, 60, 61, 62, 71, 78, 83, 88, 93, 94, 95, and 99.

Monday, November 13, 2017

Number of the day: 6674

Properties of the number 6674:

6674 = 2 × 47 × 71 is a sphenic number and squarefree.
6674 has 3 distinct prime factors, 8 divisors, 9 antidivisors and 3220 totatives.
6674 has a prime digit sum 23 in base 10.
Reversing the decimal digits of 6674 results in a semiprime.
6674 is the difference of 2 positive pentagonal numbers in 1 way.
6674 = 72 + 82 + 812 is the sum of 3 positive squares.
66742 is the sum of 3 positive squares.
6674 is a proper divisor of 2832 - 1.
6674 = '6' + '674' is the concatenation of 2 semiprime numbers.
6674 is palindromic in (at least) the following bases: 48, 93, -39, and -46.
6674 in base 36 = 55e and consists of only the digits '5' and 'e'.
6674 in base 48 = 2h2 and consists of only the digits '2' and 'h'.

The number 6674 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A137094: Numbers n such that n and the square of n use only the digits 2, 4, 5, 6 and 7.
A137380: Number of primes between (Prime[n + 1])^Pi and (Prime[n])^Pi.
A138563: Beastly fax numbers: numbers containing the fax number of the Beast (667, one more than its regular number) in their decimal expansion.
A186668: Total number of n-digit numbers requiring 11 positive biquadrates in their representation as sum of biquadrates.
A208923: Triangle of coefficients of polynomials u(n,x) jointly generated with A208908; see the Formula section.
A229014: Number of arrays of median of three adjacent elements of some length 6 0..n array, with no adjacent equal elements in the latter.
A252213: Number of (n+2)X(2+2) 0..3 arrays with every 3X3 subblock row, column, diagonal and antidiagonal sum not equal to 0 2 5 6 or 7
A252216: Number of (n+2)X(5+2) 0..3 arrays with every 3X3 subblock row, column, diagonal and antidiagonal sum not equal to 0 2 5 6 or 7
A252219: T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with every 3X3 subblock row, column, diagonal and antidiagonal sum not equal to 0 2 5 6 or 7
A260370: T(n,k)=Number of (n+2)X(k+2) 0..1 arrays with each 3X3 subblock having clockwise perimeter pattern 00000001 00000011 or 00001111

Sunday, November 12, 2017

Number of the day: 986633

Properties of the number 986633:

986633 is a cyclic number.
986633 is the 77542th prime.
986633 has 17 antidivisors and 986632 totatives.
986633 has a semiprime digit sum 35 in base 10.
Reversing the decimal digits of 986633 results in an emirp.
986633 = 4933172 - 4933162 is the difference of 2 nonnegative squares in 1 way.
986633 is the difference of 2 positive pentagonal numbers in 1 way.
986633 = 4372 + 8922 is the sum of 2 positive squares in 1 way.
986633 = 462 + 1112 + 9862 is the sum of 3 positive squares.
9866332 = 6046952 + 7796082 is the sum of 2 positive squares in 1 way.
9866332 is the sum of 3 positive squares.
986633 is a proper divisor of 376491 - 1.
986633 = '98663' + '3' is the concatenation of 2 prime numbers.
986633 = '9' + '86633' is the concatenation of 2 semiprime numbers.
986633 is an emirp in (at least) the following bases: 9, 10, 15, 21, 42, 49, 52, 59, 64, 71, 79, 83, 88, 89, 91, 92, 98, and 99.
986633 in base 42 = DDDB and consists of only the digits 'B' and 'D'.

Saturday, November 11, 2017

Number of the day: 242

Properties of the number 242:

242 = 2 × 112 is the 188th composite number and is not squarefree.
242 has 2 distinct prime factors, 6 divisors, 10 antidivisors and 110 totatives.
242 has a Fibonacci digit sum 8 in base 10.
242 has sum of divisors equal to 399 which is a sphenic number.
242 = 35 - 15 is the difference of 2 positive fifth powers in 1 way.
242 is the difference of 2 positive pentagonal numbers in 1 way.
242 = 112 + 112 is the sum of 2 positive squares in 1 way.
242 = 12 + 42 + 152 is the sum of 3 positive squares.
2422 is the sum of 3 positive squares.
242 is a proper divisor of 2412 - 1.
242 = '2' + '42' is the concatenation of 2 oblong numbers.
242 is a palindrome (in base 10).
242 is palindromic in (at least) the following bases: 3, 7, 21, -12, and -15.
242 in base 3 = 22222 and consists of only the digit '2'.
242 in base 7 = 464 and consists of only the digits '4' and '6'.
242 in base 9 = 288 and consists of only the digits '2' and '8'.
242 consists of only the digits '2' and '4'.
242 in base 11 = 200 and consists of only the digits '0' and '2'.
242 in base 15 = 112 and consists of only the digits '1' and '2'.

The number 242 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000982: a(n) = ceiling(n^2/2).
A001105: a(n) = 2*n^2.
A002088: Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010.
A002113: Palindromes in base 10.
A002822: Numbers n such that 6n-1, 6n+1 are twin primes.
A008593: Multiples of 11.
A008864: a(n) = prime(n) + 1.
A014612: Numbers that are the product of exactly three (not necessarily distinct) primes.
A028982: Squares and twice squares.
A269303: Numbers n such that (266*10^n+1)/3 is prime.

Wednesday, November 8, 2017

Number of the day: 59574745

Felix Hausdorff was born on this day 149 years ago.

Properties of the number 59574745:

59574745 is a cyclic number.
59574745 = 5 × 167 × 71347 is a sphenic number and squarefree.
59574745 has 3 distinct prime factors, 8 divisors, 11 antidivisors and 47373744 totatives.
59574745 has a semiprime digit sum 46 in base 10.
59574745 = 297873732 - 297873722 = 59574772 - 59574722 = 1784512 - 1782842 = 360912 - 352562 is the difference of 2 nonnegative squares in 4 ways.
59574745 is the sum of 2 positive triangular numbers.
59574745 is the difference of 2 positive pentagonal numbers in 4 ways.
59574745 = 3152 + 3522 + 77042 is the sum of 3 positive squares.
595747452 = 357448472 + 476597962 is the sum of 2 positive squares in 1 way.
595747452 is the sum of 3 positive squares.
59574745 is a proper divisor of 9537636 - 1.
59574745 = '5957' + '4745' is the concatenation of 2 sphenic numbers.

Tuesday, November 7, 2017

Number of the day: 32363

Properties of the number 32363:

32363 is a cyclic number.
32363 is the 3472th prime.
32363 has 15 antidivisors and 32362 totatives.
32363 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 32363 results in a semiprime.
32363 = 161822 - 161812 is the difference of 2 nonnegative squares in 1 way.
32363 is the difference of 2 positive pentagonal numbers in 1 way.
32363 = 152 + 472 + 1732 is the sum of 3 positive squares.
323632 is the sum of 3 positive squares.
32363 is a proper divisor of 3491471 - 1.
32363 is an emirp in (at least) the following bases: 11, 14, 19, 29, 31, 32, 34, 37, 52, 54, 59, 63, 66, 67, 74, 76, 77, 79, 83, 94, and 98.
32363 is palindromic in (at least) base -50.
32363 in base 33 = tnn and consists of only the digits 'n' and 't'.
32363 in base 37 = NNP and consists of only the digits 'N' and 'P'.
32363 in base 49 = DNN and consists of only the digits 'D' and 'N'.

The number 32363 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A067921: Engel expansion of sqrt(Pi/2).
A078854: Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[6, 2,6]; short d-string notation of pattern = [626].
A078959: Primes p such that the differences between the 5 consecutive primes starting with p are (6,2,6,4).
A201851: Primes of the form 7n^2 - 5.
A227282: First primes of arithmetic progressions of 7 primes each with the common difference 210.
A247197: Primes p such that 2*p^2 + 3 and 2*p^2 + 5 are also primes.
A260126: Primes that contain only the digits (2, 3, 6).
A269066: Five-digit primes whose first, third, and fifth digits are the same.
A280937: Expansion of Product_{k>=1} ((1 - x^(7*(2*k-1))) * (1 - x^(7*k)) / (1 - x^k)).

Sunday, November 5, 2017

Number of the day: 9162627

Properties of the number 9162627:

9162627 = 3 × 61 × 50069 is a sphenic number and squarefree.
9162627 has 3 distinct prime factors, 8 divisors, 19 antidivisors and 6008160 totatives.
9162627 has a semiprime digit sum 33 in base 10.
9162627 = 45813142 - 45813132 = 15271062 - 15271032 = 751342 - 750732 = 251262 - 249432 is the difference of 2 nonnegative squares in 4 ways.
9162627 is the difference of 2 positive pentagonal numbers in 2 ways.
9162627 = 732 + 1372 + 30232 is the sum of 3 positive squares.
91626272 = 61973732 + 67488002 = 16522772 + 90124202 = 34002272 + 85083602 = 48787802 + 77557232 is the sum of 2 positive squares in 4 ways.
91626272 is the sum of 3 positive squares.
9162627 is a proper divisor of 48725034 - 1.

Saturday, November 4, 2017

Number of the day: 8054397

Properties of the number 8054397:

8054397 = 34 × 13 × 7649 is the 7511217th composite number and is not squarefree.
8054397 has 3 distinct prime factors, 20 divisors, 31 antidivisors and 4955904 totatives.
8054397 has a triangular digit sum 36 in base 10.
8054397 is the difference of 2 nonnegative squares in 10 ways.
8054397 is the sum of 2 positive triangular numbers.
8054397 is the difference of 2 positive pentagonal numbers in 2 ways.
8054397 = 8462 + 27092 = 2612 + 28262 is the sum of 2 positive squares in 2 ways.
8054397 = 1162 + 1792 + 28302 is the sum of 3 positive squares.
80543972 = 16837472 + 78764402 = 14751722 + 79181552 = 45836282 + 66229652 = 30978452 + 74348282 is the sum of 2 positive squares in 4 ways.
80543972 is the sum of 3 positive squares.
8054397 is a proper divisor of 5932868 - 1.

Friday, November 3, 2017

Number of the day: 41310016

Properties of the number 41310016:

41310016 = 26 × 11 × 58679 is the 38801602th composite number and is not squarefree.
41310016 has 3 distinct prime factors, 28 divisors, 11 antidivisors and 18776960 totatives.
41310016 is the difference of 2 nonnegative squares in 10 ways.
41310016 is the difference of 2 positive pentagonal numbers in 2 ways.
41310016 = 8962 + 9842 + 62882 is the sum of 3 positive squares.
413100162 is the sum of 3 positive squares.
41310016 is a proper divisor of 35358678 - 1.

Thursday, November 2, 2017

Number of the day: 6915382

George Boole was born on this day 202 years ago.

Properties of the number 6915382:

6915382 = 2 × 661 × 5231 is a sphenic number and squarefree.
6915382 has 3 distinct prime factors, 8 divisors, 33 antidivisors and 3451800 totatives.
6915382 has a semiprime digit sum 34 in base 10.
6915382 has a Fibonacci digit sum 34 in base 10.
6915382 is the difference of 2 positive pentagonal numbers in 3 ways.
6915382 = 422 + 2932 + 26132 is the sum of 3 positive squares.
69153822 = 31386002 + 61621182 is the sum of 2 positive squares in 1 way.
69153822 is the sum of 3 positive squares.
6915382 is a proper divisor of 307660 - 1.
6915382 = '69' + '15382' is the concatenation of 2 semiprime numbers.

Wednesday, November 1, 2017

Number of the day: 540484

Properties of the number 540484:

540484 = 22 × 7 × 97 × 199 is the 495875th composite number and is not squarefree.
540484 has 4 distinct prime factors, 24 divisors, 11 antidivisors and 228096 totatives.
540484 has a semiprime digit sum 25 in base 10.
Reversing the decimal digits of 540484 results in a sphenic number.
540484 = 1351222 - 1351202 = 193102 - 192962 = 14902 - 12962 = 8782 - 4802 is the difference of 2 nonnegative squares in 4 ways.
540484 is the difference of 2 positive pentagonal numbers in 4 ways.
540484 = 722 + 1302 + 7202 is the sum of 3 positive squares.
5404842 = 3621802 + 4011842 is the sum of 2 positive squares in 1 way.
5404842 is the sum of 3 positive squares.
540484 is a proper divisor of 49112 - 1.

Tuesday, October 31, 2017

Number of the day: 128751

Karl Weierstrass was born on this day 202 years ago.

Properties of the number 128751:

128751 = 3 × 7 × 6131 is a sphenic number and squarefree.
128751 has 3 distinct prime factors, 8 divisors, 7 antidivisors and 73560 totatives.
Reversing the decimal digits of 128751 results in a sphenic number.
128751 = 643762 - 643752 = 214602 - 214572 = 92002 - 91932 = 30762 - 30552 is the difference of 2 nonnegative squares in 4 ways.
128751 is the sum of 2 positive triangular numbers.
128751 is the difference of 2 positive pentagonal numbers in 2 ways.
128751 is not the sum of 3 positive squares.
1287512 is the sum of 3 positive squares.
128751 is a proper divisor of 65910 - 1.
128751 is palindromic in (at least) the following bases: 68, and 83.
128751 in base 50 = 11P1 and consists of only the digits '1' and 'P'.

Monday, October 30, 2017

Number of the day: 96134

Properties of the number 96134:

96134 = 2 × 71 × 677 is a sphenic number and squarefree.
96134 has 3 distinct prime factors, 8 divisors, 19 antidivisors and 47320 totatives.
96134 has a prime digit sum 23 in base 10.
96134 is the difference of 2 positive pentagonal numbers in 1 way.
96134 = 132 + 222 + 3092 is the sum of 3 positive squares.
961342 = 73842 + 958502 is the sum of 2 positive squares in 1 way.
961342 is the sum of 3 positive squares.
96134 is a proper divisor of 103991 - 1.
96134 = '9' + '6134' is the concatenation of 2 semiprime numbers.
96134 is palindromic in (at least) base 58.
96134 in base 3 = 11212212112 and consists of only the digits '1' and '2'.
96134 in base 26 = 5c5c and consists of only the digits '5' and 'c'.
96134 in base 58 = SXS and consists of only the digits 'S' and 'X'.

The number 96134 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A020992: a(n) = a(n-1) + a(n-2) + a(n-3).
A145916: Even composites in A145832 with at least three distinct prime factors.
A193912: Partial sums of A193911.

Sunday, October 29, 2017

Number of the day: 1634

Properties of the number 1634:

1634 = 2 × 19 × 43 is a sphenic number and squarefree.
1634 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 756 totatives.
1634 has a semiprime digit sum 14 in base 10.
1634 has an oblong digit product 72 in base 10.
1634 is the difference of 2 positive pentagonal pyramidal numbers in 1 way.
1634 = 72 + 82 + 392 is the sum of 3 positive squares.
16342 is the sum of 3 positive squares.
1634 is a proper divisor of 12912 - 1.
1634 is palindromic in (at least) the following bases: 13, 24, 42, 85, -32, -34, and -71.
1634 in base 13 = 989 and consists of only the digits '8' and '9'.
1634 in base 16 = 662 and consists of only the digits '2' and '6'.
1634 in base 24 = 2k2 and consists of only the digits '2' and 'k'.
1634 in base 28 = 22a and consists of only the digits '2' and 'a'.

The number 1634 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002961: Numbers n such that n and n+1 have same sum of divisors.
A003321: Smallest n-th order perfect digital invariant or PDI: smallest number > 1 equal to sum of n-th powers of its digits, or 0 if no such number.
A005188: Armstrong (or Plus Perfect, or narcissistic) numbers: n-digit numbers equal to sum of n-th powers of their digits (a finite sequence, the last term being 115132219018763992565095597973971522401).
A014576: Smallest n-digit narcissistic (or Armstrong) number: smallest n-digit number equal to sum of n-th powers of its digits (or 0 if no such number exists).
A022267: a(n) = n*(9*n + 1)/2.
A023052: Powerful numbers (3): numbers n that are the sum of some fixed power of their digits.
A028557: n(n+5).
A056109: Fifth spoke of a hexagonal spiral.
A252648: Irregular table of perfect digital invariants for n > 1, i.e., numbers equal to the sum of n-th powers of their digits, read by rows.
A268261: T(n,k)=Number of length-(n+1) 0..k arrays with new repeated values introduced in sequential order starting with zero.

Saturday, October 28, 2017

Number of the day: 323866

Properties of the number 323866:

323866 = 2 × 83 × 1951 is a sphenic number and squarefree.
323866 has 3 distinct prime factors, 8 divisors, 13 antidivisors and 159900 totatives.
323866 has a triangular digit sum 28 in base 10.
Reversing the decimal digits of 323866 results in a semiprime.
323866 = 33 + 543 + 553 is the sum of 3 positive cubes in 1 way.
323866 is the sum of 2 positive triangular numbers.
323866 is the difference of 2 positive pentagonal numbers in 2 ways.
323866 = 212 + 442 + 5672 is the sum of 3 positive squares.
3238662 is the sum of 3 positive squares.
323866 is a proper divisor of 49939 - 1.
323866 = '323' + '866' is the concatenation of 2 semiprime numbers.
323866 is palindromic in (at least) the following bases: 82, 94, and -84.

The number 323866 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A114146: Number of threshold functions on n X n grid.

Friday, October 27, 2017

Number of the day: 16769

Properties of the number 16769:

16769 is a cyclic number.
16769 = 41 × 409 is semiprime and squarefree.
16769 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 16320 totatives.
16769 has a prime digit sum 29 in base 10.
Reversing the decimal digits of 16769 results in a sphenic number.
16769 = 83852 - 83842 = 2252 - 1842 is the difference of 2 nonnegative squares in 2 ways.
16769 is the sum of 2 positive triangular numbers.
16769 is the difference of 2 positive pentagonal numbers in 2 ways.
16769 = 882 + 952 = 652 + 1122 is the sum of 2 positive squares in 2 ways.
16769 = 82 + 572 + 1162 is the sum of 3 positive squares.
167692 = 83192 + 145602 = 36812 + 163602 = 12812 + 167202 = 49202 + 160312 is the sum of 2 positive squares in 4 ways.
167692 is the sum of 3 positive squares.
16769 is a proper divisor of 16378 - 1.
16769 is an emirpimes in (at least) the following bases: 4, 5, 6, 9, 19, 22, 24, 28, 35, 41, 43, 49, 51, 53, 56, 57, 59, 65, 67, 68, 69, 73, 80, 82, 83, 85, 87, 89, 90, 91, 93, and 100.
16769 is palindromic in (at least) the following bases: 38, 81, -26, and -83.
16769 in base 30 = iit and consists of only the digits 'i' and 't'.
16769 in base 38 = BNB and consists of only the digits 'B' and 'N'.

The number 16769 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A003372: Numbers that are the sum of 5 positive 7th powers.
A010013: a(0) = 1, a(n) = 23*n^2 + 2 for n>0.
A024946: s(n+3)/2, where s is A024945.
A041115: Denominators of continued fraction convergents to sqrt(66).
A041495: Denominators of continued fraction convergents to sqrt(264).
A042139: Denominators of continued fraction convergents to sqrt(594).
A239358: Number of nX5 0..2 arrays with no element equal to one plus the sum of elements to its left or one plus the sum of elements above it or one plus the sum of the elements diagonally to its northwest, modulo 3
A255967: Odd numbers n that are neither of the form p + 2^k nor of the form p - 2^k with 2^k < n, and p prime.

Thursday, October 26, 2017

Number of the day: 5462

Shiing-Shen Chern was born on this day 106 years ago.

Properties of the number 5462:

5462 = 2 × 2731 is semiprime and squarefree.
5462 has 2 distinct prime factors, 4 divisors, 17 antidivisors and 2730 totatives.
5462 has an emirp digit sum 17 in base 10.
5462 has an oblong digit product 240 in base 10.
5462 = 142 + 152 + 712 is the sum of 3 positive squares.
54622 is the sum of 3 positive squares.
5462 is a proper divisor of 68313 - 1.
5462 is an emirpimes in (at least) the following bases: 4, 7, 8, 14, 16, 18, 20, 22, 37, 38, 40, 45, 46, 68, 70, 78, 81, 82, 85, 89, 93, 94, 97, and 98.
5462 is palindromic in (at least) the following bases: 52, -4, -23, -27, -31, -60, -65, and -70.
5462 in base 4 = 1111112 and consists of only the digits '1' and '2'.
5462 in base 20 = dd2 and consists of only the digits '2' and 'd'.
5462 in base 22 = b66 and consists of only the digits '6' and 'b'.
5462 in base 26 = 822 and consists of only the digits '2' and '8'.
5462 in base 30 = 622 and consists of only the digits '2' and '6'.
5462 in base 51 = 255 and consists of only the digits '2' and '5'.
5462 in base 52 = 212 and consists of only the digits '1' and '2'.

The number 5462 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005578: a(2*n) = 2*a(2*n-1), a(2*n+1) = 2*a(2*n)-1.
A024494: C(n,1) + C(n,4) + ... + C(n,3[n/3]+1).
A047848: Array T read by diagonals; n-th difference of (T(k,n),T(k,n-1),...,T(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...
A047849: a(n) = (4^n + 2)/3.
A056105: First spoke of a hexagonal spiral.
A078008: Expansion of (1-x)/(1 - x - 2*x^2).
A131708: A024494 prefixed by a 0.
A151575: G.f.: (1+x)/(1+x-2*x^2).
A231263: T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no element unequal to a strict majority of its horizontal and antidiagonal neighbors, with values 0..2 introduced in row major order
A238555: Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 7 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=3*floor(n/2), read by rows.

Wednesday, October 25, 2017

Number of the day: 8951

Évariste Galois was born on this day 206 years ago.

Properties of the number 8951:

8951 is a cyclic number.
8951 is the 1113th prime.
8951 has 19 antidivisors and 8950 totatives.
8951 has a prime digit sum 23 in base 10.
Reversing the decimal digits of 8951 results in a sphenic number.
8951 = 44762 - 44752 is the difference of 2 nonnegative squares in 1 way.
8951 is the difference of 2 positive pentagonal numbers in 1 way.
8951 is not the sum of 3 positive squares.
89512 is the sum of 3 positive squares.
8951 is a proper divisor of 325 - 1.
8951 is an emirp in (at least) the following bases: 2, 3, 5, 9, 11, 12, 18, 20, 21, 25, 31, 34, 39, 41, 52, 59, 68, 70, 74, 75, 76, 80, 81, 82, 89, 92, 93, 94, 95, 97, and 100.
8951 is palindromic in (at least) the following bases: 33, 42, 57, and -22.
8951 in base 28 = bbj and consists of only the digits 'b' and 'j'.
8951 in base 31 = 99n and consists of only the digits '9' and 'n'.
8951 in base 32 = 8nn and consists of only the digits '8' and 'n'.
8951 in base 33 = 878 and consists of only the digits '7' and '8'.
8951 in base 41 = 5DD and consists of only the digits '5' and 'D'.
8951 in base 42 = 535 and consists of only the digits '3' and '5'.
8951 in base 54 = 33f and consists of only the digits '3' and 'f'.
8951 in base 56 = 2ll and consists of only the digits '2' and 'l'.
8951 in base 57 = 2h2 and consists of only the digits '2' and 'h'.

The number 8951 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A052283: Triangle read by rows giving numbers of directed graphs by numbers of nodes and arcs.
A060528: A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the perfect 4th, 4/3 and its complement the perfect 5th, 3/2.
A129733: List of primitive prime divisors of the numbers (3^n-1)/2 (A003462) in their order of occurrence.
A142027: Primes congruent to 23 mod 31.
A165459: Primes p such that the sum of the digits of p^2 is 16.
A189560: Least odd number k such that x' = k has n solutions, where x' is the arithmetic derivative (A003415) of x.
A196202: a(n) = 2^(prime(n)-1) mod prime(n)^2.
A237811: Primes p such that 2*p+1 and 2*p+9 are also prime.
A246342: a(0) = 12, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).
A265786: Numerators of upper primes-only best approximates (POBAs) to sqrt(5); see Comments.

Tuesday, October 24, 2017

Number of the day: 268

Properties of the number 268:

268 = 22 × 67 is the 211th composite number and is not squarefree.
268 has 2 distinct prime factors, 6 divisors, 5 antidivisors and 132 totatives.
Reversing the decimal digits of 268 results in a semiprime.
268 = 682 - 662 is the difference of 2 nonnegative squares in 1 way.
268 is the sum of 2 positive triangular numbers.
268 is the difference of 2 positive pentagonal numbers in 1 way.
268 = 62 + 62 + 142 is the sum of 3 positive squares.
2682 is the sum of 3 positive squares.
268 is a proper divisor of 4012 - 1.
268 is palindromic in (at least) the following bases: 8, 66, -14, and -89.
268 in base 8 = 414 and consists of only the digits '1' and '4'.
268 in base 11 = 224 and consists of only the digits '2' and '4'.

The number 268 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001043: Numbers that are the sum of 2 successive primes.
A002371: Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).
A002822: Numbers n such that 6n-1, 6n+1 are twin primes.
A005114: Untouchable numbers, also called nonaliquot numbers: impossible values for sum of aliquot parts of n (A001065).
A006093: a(n) = prime(n) - 1.
A014263: Numbers that contain even digits only.
A016957: a(n) = 6*n + 4.
A017113: a(n) = 8*n+4.
A054753: Numbers which are the product of a prime and the square of a different prime (p^2 * q).
A246392: Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.