Sunday, December 31, 2017

Number of the day: 8307

Carl Ludwig Siegel was born on this day 121 years ago.

Properties of the number 8307:

8307 = 32 × 13 × 71 is the 7264th composite number and is not squarefree.
8307 has 3 distinct prime factors, 12 divisors, 15 antidivisors and 5040 totatives.
8307 = 192 + … + 312 is the sum of at least 2 consecutive positive squares in 1 way.
8307 = 41542 - 41532 = 13862 - 13832 = 4662 - 4572 = 3262 - 3132 = 1262 - 872 = 942 - 232 is the difference of 2 nonnegative squares in 6 ways.
8307 is the difference of 2 positive pentagonal numbers in 1 way.
8307 = 12 + 52 + 912 is the sum of 3 positive squares.
83072 = 31952 + 76682 is the sum of 2 positive squares in 1 way.
83072 is the sum of 3 positive squares.
8307 is a proper divisor of 12794 - 1.
8307 is palindromic in (at least) the following bases: 48, 55, and -26.
8307 in base 25 = d77 and consists of only the digits '7' and 'd'.
8307 in base 45 = 44R and consists of only the digits '4' and 'R'.
8307 in base 47 = 3ZZ and consists of only the digits '3' and 'Z'.
8307 in base 48 = 3T3 and consists of only the digits '3' and 'T'.
8307 in base 52 = 33d and consists of only the digits '3' and 'd'.
8307 in base 54 = 2jj and consists of only the digits '2' and 'j'.
8307 in base 55 = 2f2 and consists of only the digits '2' and 'f'.

The number 8307 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A006875: Non-seed mu-atoms of period n in Mandelbrot set.
A022103: Fibonacci sequence beginning 1, 13.
A031431: Least term in period of continued fraction for sqrt(n) is 7.
A056078: Number of proper T_1-hypergraphs with 3 labeled nodes and n hyperedges.
A136376: a(n) = n*F(n) + (n-1)*F(n-1).
A157365: 49n^2 + 2n.
A218563: Numbers n such that n^2 + 1 is divisible by a 4th power.
A218564: Numbers n such that n^2 + 1 is divisible by a 5th power.
A241744: Number of partitions p of n such that (number of numbers in p of form 3k) = (number of numbers in p of form 3k+1).
A252407: T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with every 3X3 subblock row and diagonal sum equal to 2 3 4 6 or 7 and every 3X3 column and antidiagonal sum not equal to 2 3 4 6 or 7

Saturday, December 30, 2017

Number of the day: 93149

Properties of the number 93149:

93149 = 72 × 1901 is the 84151th composite number and is not squarefree.
93149 has 2 distinct prime factors, 6 divisors, 7 antidivisors and 79800 totatives.
93149 has an emirpimes digit sum 26 in base 10.
Reversing the decimal digits of 93149 results in a semiprime.
93149 = 465752 - 465742 = 66572 - 66502 = 9752 - 9262 is the difference of 2 nonnegative squares in 3 ways.
93149 is the difference of 2 positive pentagonal numbers in 3 ways.
93149 = 1822 + 2452 is the sum of 2 positive squares in 1 way.
93149 = 22 + 272 + 3042 is the sum of 3 positive squares.
931492 = 269012 + 891802 is the sum of 2 positive squares in 1 way.
931492 is the sum of 3 positive squares.
93149 is a proper divisor of 88395 - 1.
93149 = '9' + '3149' is the concatenation of 2 semiprime numbers.
93149 is palindromic in (at least) the following bases: 47, and -62.
93149 in base 47 = g7g and consists of only the digits '7' and 'g'.
93149 in base 61 = P22 and consists of only the digits '2' and 'P'.

Friday, December 29, 2017

Number of the day: 791281

Properties of the number 791281:

791281 is a cyclic number.
791281 = 647 × 1223 is semiprime and squarefree.
791281 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 789412 totatives.
791281 has a triangular digit sum 28 in base 10.
Reversing the decimal digits of 791281 results in an emirpimes.
791281 = 253 + 363 + 903 is the sum of 3 positive cubes in 1 way.
791281 = 3956412 - 3956402 = 9352 - 2882 is the difference of 2 nonnegative squares in 2 ways.
791281 is the sum of 2 positive triangular numbers.
791281 is the difference of 2 positive pentagonal numbers in 1 way.
791281 = 1092 + 1502 + 8702 is the sum of 3 positive squares.
7912812 is the sum of 3 positive squares.
791281 is a proper divisor of 1791598 - 1.
791281 is an emirpimes in (at least) the following bases: 2, 3, 6, 10, 13, 15, 23, 24, 25, 26, 27, 30, 31, 33, 39, 41, 43, 44, 46, 49, 50, 54, 61, 63, 69, 76, 81, 84, 85, 88, 92, 93, 94, 95, 96, 98, and 99.
791281 is palindromic in (at least) base 28.
791281 in base 28 = 18181 and consists of only the digits '1' and '8'.

Thursday, December 28, 2017

Number of the day: 47643

John von Neumann was born on this day 114 years ago.

Properties of the number 47643:

47643 is a cyclic number.
47643 = 3 × 15881 is semiprime and squarefree.
47643 has 2 distinct prime factors, 4 divisors, 17 antidivisors and 31760 totatives.
47643 has a triangular digit product 2016 in base 10.
Reversing the decimal digits of 47643 results in a sphenic number.
47643 = 238222 - 238212 = 79422 - 79392 is the difference of 2 nonnegative squares in 2 ways.
47643 is the sum of 2 positive triangular numbers.
47643 is the difference of 2 positive pentagonal numbers in 1 way.
47643 = 52 + 232 + 2172 is the sum of 3 positive squares.
476432 = 120002 + 461072 is the sum of 2 positive squares in 1 way.
476432 is the sum of 3 positive squares.
47643 is a proper divisor of 193340 - 1.
47643 = '47' + '643' is the concatenation of 2 prime numbers.
47643 is an emirpimes in (at least) the following bases: 3, 6, 8, 9, 15, 16, 21, 33, 34, 37, 41, 47, 51, 53, 60, 69, 78, 80, 81, 89, 96, and 97.
47643 in base 45 = NNX and consists of only the digits 'N' and 'X'.

The number 47643 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A157443: 121n^2 - 38n + 3.
A188475: (2*n^3 + 3*n^2 + n + 3)/3.
A224257: Number of nX3 0..2 arrays with rows and antidiagonals unimodal and columns nondecreasing

Wednesday, December 27, 2017

Number of the day: 956122

Johannes Kepler was born on this day 446 years ago.

Johannes Kepler was born on this day 446 years ago.

Jacob Bernoulli was born on this day 363 years ago.

Properties of the number 956122:

956122 = 2 × 193 × 2477 is a sphenic number and squarefree.
956122 has 3 distinct prime factors, 8 divisors, 23 antidivisors and 475392 totatives.
956122 has a semiprime digit sum 25 in base 10.
Reversing the decimal digits of 956122 results in a prime.
956122 is the difference of 2 positive pentagonal numbers in 4 ways.
956122 = 5912 + 7792 = 1312 + 9692 is the sum of 2 positive squares in 2 ways.
956122 = 232 + 1682 + 9632 is the sum of 3 positive squares.
9561222 = 4706302 + 8322722 = 2575602 + 9207782 = 2538782 + 9218002 = 6747282 + 6774302 is the sum of 2 positive squares in 4 ways.
9561222 is the sum of 3 positive squares.
956122 is a proper divisor of 1931619 - 1.
956122 = '95' + '6122' is the concatenation of 2 semiprime numbers.

Sunday, December 24, 2017

Number of the day: 563497217

Charles Hermite was born on this day 195 years ago.

Properties of the number 563497217:

563497217 is a cyclic number.
563497217 = 23 × 24499879 is semiprime and squarefree.
563497217 has 2 distinct prime factors, 4 divisors, 31 antidivisors and 538997316 totatives.
Reversing the decimal digits of 563497217 results in an emirpimes.
563497217 = 2817486092 - 2817486082 = 122499512 - 122499282 is the difference of 2 nonnegative squares in 2 ways.
563497217 is the difference of 2 positive pentagonal numbers in 2 ways.
563497217 = 342 + 6192 + 237302 is the sum of 3 positive squares.
5634972172 is the sum of 3 positive squares.
563497217 is a proper divisor of 7611551 - 1.
563497217 = '56349721' + '7' is the concatenation of 2 prime numbers.
563497217 is an emirpimes in (at least) the following bases: 2, 5, 6, 10, 12, 13, 17, 19, 21, 23, 29, 33, 37, 39, 40, 48, 49, 50, 51, 57, 60, 61, 64, 65, 66, 67, 70, 83, 84, 89, 91, 94, and 97.

Saturday, December 23, 2017

Number of the day: 97548

Properties of the number 97548:

97548 = 22 × 3 × 11 × 739 is the 88168th composite number and is not squarefree.
97548 has 4 distinct prime factors, 24 divisors, 15 antidivisors and 29520 totatives.
97548 has a semiprime digit sum 33 in base 10.
97548 = 243882 - 243862 = 81322 - 81262 = 22282 - 22062 = 7722 - 7062 is the difference of 2 nonnegative squares in 4 ways.
97548 is the sum of 2 positive triangular numbers.
97548 is the difference of 2 positive pentagonal numbers in 2 ways.
97548 = 22 + 382 + 3102 is the sum of 3 positive squares.
975482 is the sum of 3 positive squares.
97548 is a proper divisor of 4196 - 1.
97548 is palindromic in (at least) the following bases: 63, and -55.
97548 in base 54 = XOO and consists of only the digits 'O' and 'X'.

The number 97548 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A047003: T(n,n-1), array T given by A047000.
A149915: Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, -1, 0), (-1, 0, 1), (-1, 1, -1), (0, 0, 1), (1, 0, 0)}
A292346: The forgotten topological index of the Aztec diamond AZ(n) (see the Ramanes et al. reference, Theorem 2.1).

Friday, December 22, 2017

Number of the day: 301992

Srinivasa Ramanujan was born on this day 130 years ago.

Properties of the number 301992:

301992 = 23 × 3 × 12583 is the 275829th composite number and is not squarefree.
301992 has 3 distinct prime factors, 16 divisors, 15 antidivisors and 100656 totatives.
Reversing the decimal digits of 301992 results in a sphenic number.
301992 = 754992 - 754972 = 377512 - 377472 = 251692 - 251632 = 125892 - 125772 is the difference of 2 nonnegative squares in 4 ways.
301992 = 22 + 1122 + 5382 is the sum of 3 positive squares.
3019922 is the sum of 3 positive squares.
301992 is a proper divisor of 16976 - 1.

Thursday, December 21, 2017

Number of the day: 8733

Properties of the number 8733:

8733 is a cyclic number.
8733 = 3 × 41 × 71 is a sphenic number and squarefree.
8733 has 3 distinct prime factors, 8 divisors, 13 antidivisors and 5600 totatives.
8733 has a semiprime digit sum 21 in base 10.
8733 has a Fibonacci digit sum 21 in base 10.
8733 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 8733 results in a sphenic number.
8733 = 43672 - 43662 = 14572 - 14542 = 1272 - 862 = 972 - 262 is the difference of 2 nonnegative squares in 4 ways.
8733 is the sum of 2 positive triangular numbers.
8733 is the difference of 2 positive pentagonal numbers in 1 way.
8733 = 102 + 132 + 922 is the sum of 3 positive squares.
87332 = 19172 + 85202 is the sum of 2 positive squares in 1 way.
87332 is the sum of 3 positive squares.
8733 is a proper divisor of 8775 - 1.
8733 = '87' + '33' is the concatenation of 2 semiprime numbers.
8733 is palindromic in (at least) the following bases: 23, 43, 74, and -27.
8733 in base 23 = gbg and consists of only the digits 'b' and 'g'.
8733 in base 26 = cnn and consists of only the digits 'c' and 'n'.
8733 in base 42 = 4dd and consists of only the digits '4' and 'd'.
8733 in base 43 = 4V4 and consists of only the digits '4' and 'V'.

The number 8733 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005320: a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.
A041016: Numerators of continued fraction convergents to sqrt(12).
A131039: Expansion of (1-x)*(2*x^2-4*x+1)/(1-2*x+5*x^2-4*x^3+x^4).
A143642: Numerators of principal and intermediate convergents to 3^(1/2).
A143643: Lower principal and intermediate convergents to 3^(1/2).
A194480: T(n,k) = number of ways to arrange k indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.
A212064: Number of (w,x,y,z) with all terms in {1,...,n} and w^2>=x*y*z.
A227418: Array A(n,k) with all numbers m such that 3*m^2 +- 3^k is a square and their corresponding square roots, read downward by diagonals.
A229362: a(n) = n for n = 1, 2, 3; for n > 3: a(n) = number of partitions of n into preceding terms.
A259593: Numerators of the other-side convergents to sqrt(3).

Tuesday, December 19, 2017

Number of the day: 2298

Properties of the number 2298:

2298 = 2 × 3 × 383 is a sphenic number and squarefree.
2298 has 3 distinct prime factors, 8 divisors, 5 antidivisors and 764 totatives.
2298 has a semiprime digit sum 21 in base 10.
2298 has a Fibonacci digit sum 21 in base 10.
2298 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 2298 results in a sphenic number.
2298 is the sum of 2 positive triangular numbers.
2298 is the difference of 2 positive pentagonal numbers in 2 ways.
2298 = 52 + 82 + 472 is the sum of 3 positive squares.
22982 is the sum of 3 positive squares.
2298 is a proper divisor of 15312 - 1.
2298 is palindromic in (at least) the following bases: 27, 28, and -41.
2298 in base 3 = 10011010 and consists of only the digits '0' and '1'.
2298 in base 9 = 3133 and consists of only the digits '1' and '3'.
2298 in base 15 = a33 and consists of only the digits '3' and 'a'.
2298 in base 19 = 66i and consists of only the digits '6' and 'i'.
2298 in base 26 = 3aa and consists of only the digits '3' and 'a'.
2298 in base 27 = 343 and consists of only the digits '3' and '4'.
2298 in base 28 = 2q2 and consists of only the digits '2' and 'q'.
2298 in base 47 = 11g and consists of only the digits '1' and 'g'.

The number 2298 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A028878: a(n) = (n+3)^2 - 6.
A051228: Bernoulli number B_{n} has denominator 42.
A083096: Numbers n such that 3 divides sum(k=1,n, C(2k,k) ).
A084785: Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784).
A090801: List of distinct numbers appearing as denominators of Bernoulli numbers.
A217729: Trajectory of 40 under the map n-> A006369(n).
A218038: Numbers n such that Q(sqrt(n)) has class number 6.
A225196: Number of 6-line partitions of n (i.e., planar partitions of n with at most 6 lines).
A269606: T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by one or less.
A270850: T(n,k)=Number of nXnXn triangular 0..k arrays with some element plus some adjacent element totalling k+1, k or k-1 exactly once.

Monday, December 18, 2017

Number of the day: 91686001

Properties of the number 91686001:

91686001 is a cyclic number.
91686001 = 11 × 19 × 193 × 2273 is the 86376850th composite number and is squarefree.
91686001 has 4 distinct prime factors, 16 divisors, 23 antidivisors and 78520320 totatives.
91686001 has an emirp digit sum 31 in base 10.
Reversing the decimal digits of 91686001 results in a sphenic number.
91686001 = 458430012 - 458430002 = 41675512 - 41675402 = 24127992 - 24127802 = 2376252 - 2374322 = 2194492 - 2192402 = 226552 - 205322 = 213052 - 190322 = 143352 - 106682 is the difference of 2 nonnegative squares in 8 ways.
91686001 is the sum of 2 positive triangular numbers.
91686001 is the difference of 2 positive pentagonal numbers in 8 ways.
91686001 = 3692 + 5562 + 95522 is the sum of 3 positive squares.
916860012 = 603842802 + 689931992 = 451304152 + 798095762 = 161847512 + 902462002 = 303334242 + 865228652 is the sum of 2 positive squares in 4 ways.
916860012 is the sum of 3 positive squares.
91686001 is a proper divisor of 1493960 - 1.
91686001 = '9' + '1686001' is the concatenation of 2 semiprime numbers.

Sunday, December 17, 2017

Number of the day: 61348

Marius Sophus Lie was born on this day 175 years ago.

Properties of the number 61348:

61348 = 22 × 72 × 313 is the 55175th composite number and is not squarefree.
61348 has 3 distinct prime factors, 18 divisors, 15 antidivisors and 26208 totatives.
61348 has a semiprime digit sum 22 in base 10.
61348 = 153382 - 153362 = 21982 - 21842 = 3622 - 2642 is the difference of 2 nonnegative squares in 3 ways.
61348 is the difference of 2 positive pentagonal numbers in 3 ways.
61348 = 1682 + 1822 is the sum of 2 positive squares in 1 way.
61348 = 382 + 482 + 2402 is the sum of 3 positive squares.
613482 = 49002 + 611522 is the sum of 2 positive squares in 1 way.
613482 is the sum of 3 positive squares.
61348 is a proper divisor of 187714 - 1.
61348 is palindromic in (at least) base 5.
61348 in base 3 = 10010011011 and consists of only the digits '0' and '1'.
61348 in base 25 = 3n3n and consists of only the digits '3' and 'n'.
61348 in base 27 = 3344 and consists of only the digits '3' and '4'.
61348 in base 39 = 11D1 and consists of only the digits '1' and 'D'.
61348 in base 45 = UDD and consists of only the digits 'D' and 'U'.

The number 61348 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A027105: a(n) =Sum{T(n,k)*T(n,2n-k)}, 0<=k<=n-1, T given by A027082.
A217631: Number of nX2 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX2 array
A241431: Number of nX4 0..3 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 zero plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4
A276492: Numbers n such that 5*10^n + 59 is prime.

Saturday, December 16, 2017

Number of the day: 875875

Properties of the number 875875:

875875 = 53 × 72 × 11 × 13 is the 806354th composite number and is not squarefree.
875875 has 4 distinct prime factors, 48 divisors, 59 antidivisors and 504000 totatives.
875875 has sum of divisors equal to 1493856 which is a triangular number.
875875 is the difference of 2 nonnegative squares in 24 ways.
875875 is the difference of 2 positive pentagonal numbers in 21 ways.
875875 = 252 + 692 + 9332 is the sum of 3 positive squares.
8758752 = 3368752 + 8085002 = 2156002 + 8489252 = 4446752 + 7546002 = 5497802 + 6818352 = 970202 + 8704852 = 307232 + 8753362 = 5999072 + 6381762 = 5255252 + 7007002 = 2452452 + 8408402 = 3083082 + 8198192 is the sum of 2 positive squares in 10 ways.
8758752 is the sum of 3 positive squares.
875875 is a proper divisor of 30728 - 1.
875875 = '87587' + '5' is the concatenation of 2 prime numbers.

The number 875875 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A100646: Denominator of Cotesian number C(n,2).
A160555: Denominator of Laguerre(n, -12).
A160608: Denominator of Laguerre(n, -6).
A160632: Denominator of Laguerre(n, 6).
A160672: Denominator of Laguerre(n, 12).
A185505: a(n) = (7*n^4 + 5*n^2)/12.

Friday, December 15, 2017

Number of the day: 4286

János Bolyai was born on this day 215 years ago.

Properties of the number 4286:

4286 = 2 × 2143 is semiprime and squarefree.
4286 has 2 distinct prime factors, 4 divisors, 3 antidivisors and 2142 totatives.
4286 has an oblong digit sum 20 in base 10.
4286 = 52 + 62 + 652 is the sum of 3 positive squares.
42862 is the sum of 3 positive squares.
4286 is a proper divisor of 3493 - 1.
4286 is an emirpimes in (at least) the following bases: 9, 13, 14, 15, 27, 28, 34, 35, 36, 43, 44, 45, 52, 53, 55, 58, 59, 62, 63, 64, 69, 71, 74, 81, 84, 86, 89, 91, and 98.
4286 is palindromic in (at least) the following bases: 19, 23, 42, -51, and -63.
4286 in base 17 = ee2 and consists of only the digits '2' and 'e'.
4286 in base 19 = bgb and consists of only the digits 'b' and 'g'.
4286 in base 22 = 8ii and consists of only the digits '8' and 'i'.
4286 in base 23 = 828 and consists of only the digits '2' and '8'.
4286 in base 41 = 2MM and consists of only the digits '2' and 'M'.
4286 in base 42 = 2I2 and consists of only the digits '2' and 'I'.

The number 4286 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A070143: Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer area, having relatively prime side lengths.
A070209: Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer inradius.
A075277: Interprimes (A024675) which are of the form s*prime, s=2.
A100229: Triangle, read by rows, of the coefficients of [x^k] in G100228(x)^n such that the row sums are 4^n-1 for n>0, where G100228(x) is the g.f. of A100228.
A100230: Main diagonal of triangle A100229.
A111570: a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4.
A196087: Sum of all parts minus the total numbers of parts of all partitions of n.
A248201: Numbers n such that n-1, n and n+1 are all squarefree semiprimes.
A276743: G.f.: Sum_{n>=0} [ Sum_{k>=1} k^n * x^k ]^n.
A285212: Expansion of Product_{k>=0} (1-x^(3*k+2))^(3*k+2).

Thursday, December 14, 2017

Number of the day: 748724

Properties of the number 748724:

748724 = 22 × 187181 is the 688580th composite number and is not squarefree.
748724 has 2 distinct prime factors, 6 divisors, 43 antidivisors and 374360 totatives.
Reversing the decimal digits of 748724 results in a semiprime.
748724 = 1871822 - 1871802 is the difference of 2 nonnegative squares in 1 way.
748724 is the difference of 2 positive pentagonal numbers in 1 way.
748724 = 5502 + 6682 is the sum of 2 positive squares in 1 way.
748724 = 282 + 382 + 8642 is the sum of 3 positive squares.
7487242 = 1437242 + 7348002 is the sum of 2 positive squares in 1 way.
7487242 is the sum of 3 positive squares.
748724 is a proper divisor of 59196 - 1.

Wednesday, December 13, 2017

Number of the day: 380653

George Pólya was born on this day 130 years ago.

Properties of the number 380653:

380653 is a cyclic number.
380653 = 7 × 13 × 47 × 89 is the 348304th composite number and is squarefree.
380653 has 4 distinct prime factors, 16 divisors, 23 antidivisors and 291456 totatives.
380653 has a semiprime digit sum 25 in base 10.
380653 = 1903272 - 1903262 = 271932 - 271862 = 146472 - 146342 = 40732 - 40262 = 21832 - 20942 = 21372 - 20462 = 7432 - 4142 = 6172 - 62 is the difference of 2 nonnegative squares in 8 ways.
380653 is the difference of 2 positive pentagonal numbers in 6 ways.
380653 = 512 + 1042 + 6062 is the sum of 3 positive squares.
3806532 = 1668032 + 3421602 = 223722 + 3799952 = 2516852 + 2855722 = 1464052 + 3513722 is the sum of 2 positive squares in 4 ways.
3806532 is the sum of 3 positive squares.
380653 is a proper divisor of 28366 - 1.
380653 is palindromic in (at least) base 46.
380653 in base 34 = 9n9n and consists of only the digits '9' and 'n'.
380653 in base 46 = 3ff3 and consists of only the digits '3' and 'f'.

The number 380653 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A287583: Number of set partitions of [n] such that all absolute differences between least elements of consecutive blocks and between consecutive elements within the blocks are not larger than four.

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].