Wednesday, November 30, 2016

Number of the day: 696557958264193

Properties of the number 696557958264193:

696557958264193 = 67 × 159857 × 65035547 is a sphenic number and squarefree.
696557958264193 has 3 distinct prime factors, 8 divisors, 73 antidivisors and 686157267930816 totatives.
696557958264193 has an emirpimes digit sum 85 in base 10.
696557958264193 = 3482789791320972 - 3482789791320962 = 51981937184232 - 51981937183562 = 21787707532 - 21786108962 = 378729832 - 271625642 is the difference of 2 nonnegative squares in 4 ways.
696557958264193 is the sum of 2 positive triangular numbers.
696557958264193 is the difference of 2 positive pentagonal numbers in 4 ways.
696557958264193 = 702 + 344432 + 263923622 is the sum of 3 positive squares.
6965579582641932 = 4266094503453452 + 5506336042208322 is the sum of 2 positive squares in 1 way.
6965579582641932 is the sum of 3 positive squares.
696557958264193 is a divisor of 1847160767869712 - 1.
696557958264193 = '69' + '6557958264193' is the concatenation of 2 semiprime numbers.

Tuesday, November 29, 2016

Number of the day: 96075

Properties of the number 96075:

96075 = 32 × 52 × 7 × 61 is the 86816th composite number and is not squarefree.
96075 has 4 distinct prime factors, 36 divisors, 41 antidivisors and 43200 totatives.
96075 is the difference of 2 nonnegative squares in 18 ways.
96075 is the sum of 2 positive triangular numbers.
96075 is the difference of 2 positive pentagonal numbers in 4 ways.
96075 = 52 + 1012 + 2932 is the sum of 3 positive squares.
960752 = 428402 + 859952 = 173252 + 945002 = 430922 + 858692 = 652052 + 705602 = 98282 + 955712 = 576452 + 768602 = 269012 + 922322 is the sum of 2 positive squares in 7 ways.
960752 is the sum of 3 positive squares.
96075 is a divisor of 10516 - 1.
96075 is palindromic in (at least) the following bases: -57, and -86.
96075 in base 7 = 550050 and consists of only the digits '0' and '5'.
96075 in base 56 = UZZ and consists of only the digits 'U' and 'Z'.

Monday, November 28, 2016

Number of the day: 5800

Properties of the number 5800:

5800 = 23 × 52 × 29 is the 5039th composite number and is not squarefree.
5800 has 3 distinct prime factors, 24 divisors, 11 antidivisors and 2240 totatives.
5800 has an emirp digit sum 13 in base 10.
5800 has a Fibonacci digit sum 13 in base 10.
5800 = 14512 - 14492 = 7272 - 7232 = 2952 - 2852 = 1552 - 1352 = 832 - 332 = 792 - 212 is the difference of 2 nonnegative squares in 6 ways.
5800 is the sum of 2 positive triangular numbers.
5800 is the difference of 2 positive pentagonal numbers in 3 ways.
5800 = 302 + 702 = 382 + 662 = 182 + 742 is the sum of 2 positive squares in 3 ways.
5800 = 302 + 422 + 562 is the sum of 3 positive squares.
58002 = 34802 + 46402 = 9602 + 57202 = 6802 + 57602 = 40002 + 42002 = 29122 + 50162 = 16242 + 55682 = 26642 + 51522 is the sum of 2 positive squares in 7 ways.
58002 is the sum of 3 positive squares.
5800 is a divisor of 3492 - 1.
5800 is palindromic in (at least) the following bases: 36, 99, and -42.
5800 in base 3 = 21221211 and consists of only the digits '1' and '2'.
5800 in base 12 = 3434 and consists of only the digits '3' and '4'.
5800 in base 17 = 1313 and consists of only the digits '1' and '3'.
5800 in base 35 = 4pp and consists of only the digits '4' and 'p'.
5800 in base 36 = 4h4 and consists of only the digits '4' and 'h'.

The number 5800 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A026054: dot_product(n,n-1,...2,1).(3,4,...,n,1,2).
A060488: Number of 4-block ordered tricoverings of an unlabeled n-set.
A082227: Main diagonal of array A082224.
A138992: a(n) = Frobenius number for 6 successive primes = F[p(n),p(n+1),p(n+2),p(n+3),p(n+4),p(n+5)].
A164770: Numbers n with property that average digit of n^2 is 2.
A184071: T(n,k)=Number of (n+1)X(k+1) binary arrays with rows and columns in nondecreasing order and with no 2X2 subblock sum differing from a horizontal or vertical neighbor subblock sum by more than one
A211971: Column 0 of square array A211970 (in which column 1 is A000041).
A216046: Expansion of (chi(-x) / chi^3(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.
A241343: Number of partitions p of n such that neither floor(mean(p)) nor ceiling(mean(p)) is a part.
A241515: Number of partitions of n such that (number parts having multiplicity 1) is a part or (number of parts > 1) is a part.

Sunday, November 27, 2016

Number of the day: 62389

Properties of the number 62389:

62389 = 89 × 701 is semiprime and squarefree.
62389 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 61600 totatives.
62389 has a triangular digit sum 28 in base 10.
Reversing the decimal digits of 62389 results in a sphenic number.
62389 = 311952 - 311942 = 3952 - 3062 is the difference of 2 nonnegative squares in 2 ways.
62389 is the difference of 2 positive pentagonal numbers in 2 ways.
62389 = 1702 + 1832 = 902 + 2332 is the sum of 2 positive squares in 2 ways.
62389 = 282 + 332 + 2462 is the sum of 3 positive squares.
623892 = 419402 + 461892 = 273392 + 560802 = 45892 + 622202 = 231402 + 579392 is the sum of 2 positive squares in 4 ways.
623892 is the sum of 3 positive squares.
62389 is a divisor of 16777 - 1.
62389 = '6238' + '9' is the concatenation of 2 semiprime numbers.
62389 is an emirpimes in (at least) the following bases: 3, 4, 5, 8, 9, 12, 15, 17, 18, 21, 22, 23, 25, 26, 27, 31, 32, 34, 38, 50, 53, 56, 60, 64, 70, 74, 76, 82, and 98.
62389 is palindromic in (at least) the following bases: 13, 69, -18, and -62.
62389 in base 13 = 22522 and consists of only the digits '2' and '5'.
62389 in base 54 = LLJ and consists of only the digits 'J' and 'L'.

Saturday, November 26, 2016

Number of the day: 674714844174

Norbert Wiener was born on this day 122 years ago.

Properties of the number 674714844174:

674714844174 = 2 × 3 × 7 × 41 × 53 × 7392839 is composite and squarefree.
674714844174 has 6 distinct prime factors, 64 divisors, 43 antidivisors and 184525236480 totatives.
674714844174 has a semiprime digit sum 57 in base 10.
674714844174 is the difference of 2 positive pentagonal numbers in 8 ways.
674714844174 = 4102 + 49932 + 8213952 is the sum of 3 positive squares.
6747148441742 = 4735113379502 + 4806528204242 = 1481081365262 + 6582583845602 = 2220069551702 + 6371444363762 = 3564531252242 + 5728710941102 is the sum of 2 positive squares in 4 ways.
6747148441742 is the sum of 3 positive squares.
674714844174 is a divisor of 8314785676 - 1.
674714844174 = '67471' + '4844174' is the concatenation of 2 semiprime numbers.

Friday, November 25, 2016

Number of the day: 3950558

Properties of the number 3950558:

3950558 = 2 × 1975279 is semiprime and squarefree.
3950558 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 1975278 totatives.
3950558 has a semiprime digit sum 35 in base 10.
Reversing the decimal digits of 3950558 results in an emirpimes.
3950558 = 252 + 422 + 19872 is the sum of 3 positive squares.
39505582 is the sum of 3 positive squares.
3950558 is a divisor of 10717327 - 1.
3950558 = '39505' + '58' is the concatenation of 2 semiprime numbers.
3950558 is an emirpimes in (at least) the following bases: 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 19, 20, 23, 34, 42, 43, 44, 48, 50, 54, 56, 61, 62, 64, 66, 70, 80, 85, 90, 92, 93, and 99.

Thursday, November 24, 2016

Number of the day: 87945048061577

Properties of the number 87945048061577:

87945048061577 = 7 × 19 × 4673 × 141502453 is composite and squarefree.
87945048061577 has 4 distinct prime factors, 16 divisors, 35 antidivisors and 71398741220352 totatives.
87945048061577 has an emirp digit sum 71 in base 10.
Reversing the decimal digits of 87945048061577 results in a semiprime.
87945048061577 = 439725240307892 - 439725240307882 = 62817891472592 - 62817891472522 = 23143433700512 - 23143433700322 = 3306204815012 - 3306204813682 = 94099154612 - 94099107882 = 13442896592 - 13442569482 = 4953029792 - 4952141922 = 710619812 - 704404722 is the difference of 2 nonnegative squares in 8 ways.
87945048061577 is the difference of 2 positive pentagonal numbers in 8 ways.
87945048061577 = 75682 + 144032 + 93778882 is the sum of 3 positive squares.
879450480615772 = 460341476870732 + 749345629551402 = 179164745890482 + 861007050891752 = 503513790302602 + 721045775821772 = 603346735719352 + 639848313549482 is the sum of 2 positive squares in 4 ways.
879450480615772 is the sum of 3 positive squares.
87945048061577 is a divisor of 691794590692 - 1.
87945048061577 = '87' + '945048061577' is the concatenation of 2 semiprime numbers.

Wednesday, November 23, 2016

Number of the day: 11232016

John Wallis was born on this day 400 years ago.

Happy Fibonacci Day!

Properties of the number 11232016:

11232016 = 24 × 37 × 18973 is the 10491196th composite number and is not squarefree.
11232016 has 3 distinct prime factors, 20 divisors, 15 antidivisors and 5463936 totatives.
Reversing the decimal digits of 11232016 results in a semiprime.
11232016 = 28080052 - 28080032 = 14040042 - 14040002 = 7020052 - 7019972 = 759292 - 758552 = 380202 - 378722 = 191212 - 188252 is the difference of 2 nonnegative squares in 6 ways.
11232016 is the difference of 2 positive pentagonal numbers in 2 ways.
11232016 = 19802 + 27042 = 9962 + 32002 is the sum of 2 positive squares in 2 ways.
11232016 = 2562 + 4442 + 33122 is the sum of 3 positive squares.
112320162 = 36428162 + 106248802 = 33912162 + 107078402 = 63744002 + 92479842 = 66807202 + 90291842 is the sum of 2 positive squares in 4 ways.
112320162 is the sum of 3 positive squares.
11232016 is a divisor of 47306 - 1.
11232016 = '1123201' + '6' is the concatenation of 2 semiprime numbers.

Tuesday, November 22, 2016

Number of the day: 706

Properties of the number 706:

706 = 2 × 353 is semiprime and squarefree.
706 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 352 totatives.
706 has an emirp digit sum 13 in base 10.
706 has a Fibonacci digit sum 13 in base 10.
Reversing the decimal digits of 706 results in a prime.
706 is the sum of 2 positive triangular numbers.
706 is the difference of 2 positive pentagonal numbers in 2 ways.
706 is the difference of 2 positive pentagonal pyramidal numbers in 1 way.
706 = 92 + 252 is the sum of 2 positive squares in 1 way.
706 = 152 + 152 + 162 is the sum of 3 positive squares.
7062 = 4502 + 5442 is the sum of 2 positive squares in 1 way.
7062 is the sum of 3 positive squares.
706 is a divisor of 3114 - 1.
706 is an emirpimes in (at least) the following bases: 4, 9, 11, 17, 20, 22, 25, 27, 29, 31, 32, 40, 41, 47, 48, 50, 51, 53, 60, 66, 69, 71, 76, 78, 79, 83, 85, 87, 89, 90, 95, 98, and 99.
706 is palindromic in (at least) the following bases: 13, 16, -22, and -47.
706 in base 12 = 4aa and consists of only the digits '4' and 'a'.
706 in base 13 = 424 and consists of only the digits '2' and '4'.
706 in base 16 = 2c2 and consists of only the digits '2' and 'c'.
706 in base 26 = 114 and consists of only the digits '1' and '4'.

The number 706 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000698: A problem of configurations: a(0) = 1; for n>0, a(n) = (2n-1)!! - Sum_{k=1..n-1} (2k-1)!! a(n-k). Also the number of shellings of an n-cube, divided by 2^n n!.
A003336: Numbers that are the sum of 2 nonzero 4th powers.
A006753: Smith (or joke) numbers: composite numbers n such that sum of digits of n = sum of digits of prime factors of n (counted with multiplicity).
A051225: Bernoulli number B_{2n} has denominator 30.
A052329: Number of rooted trees with a forbidden limb of length 6.
A055013: Sum of 4th powers of digits of n.
A056109: Fifth spoke of a hexagonal spiral.
A100037: Positions of occurrences of the natural numbers as second subsequence in A100035.
A181568: Numbers n such that the largest prime factor of n^2-1 is 101.
A213249: Triangle T(n,k) of numbers of distinct shapes under rotation of non-extendable (complete) non-self-adjacent simple paths within a square lattice bounded by rectangles with nodal dimensions n and k, n >= k >= 2.

Monday, November 21, 2016

Number of the day: 4894

Properties of the number 4894:

4894 = 2 × 2447 is semiprime and squarefree.
4894 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 2446 totatives.
4894 has a semiprime digit sum 25 in base 10.
4894 is the sum of 2 positive triangular numbers.
4894 is the difference of 2 positive pentagonal numbers in 2 ways.
4894 = 232 + 422 + 512 is the sum of 3 positive squares.
48942 is the sum of 3 positive squares.
4894 is a divisor of 31223 - 1.
4894 = '489' + '4' is the concatenation of 2 semiprime numbers.
4894 is an emirpimes in (at least) the following bases: 2, 3, 4, 5, 8, 14, 19, 29, 31, 33, 34, 36, 41, 43, 47, 48, 49, 50, 52, 61, 62, 64, 68, 71, 78, 84, 85, 89, 91, and 99.
4894 is palindromic in (at least) the following bases: 22, -18, and -27.
4894 in base 17 = gff and consists of only the digits 'f' and 'g'.
4894 in base 22 = a2a and consists of only the digits '2' and 'a'.
4894 in base 26 = 766 and consists of only the digits '6' and '7'.
4894 in base 28 = 66m and consists of only the digits '6' and 'm'.

The number 4894 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A006227: Number of n-dimensional space groups (including enantiomorphs).
A014126: Number of partitions of 2*n into at most 4 parts.
A021664: Expansion of 1/((1-x)(1-3x)(1-8x)(1-11x)).
A089508: Solution to a binomial problem together with companion sequence A081016(n-1).
A111569: a(n) = a(n-1) + a(n-3) + a(n-4) for n>3, a(0) = -1, a(1) = 1, a(2) = 2, a(3) = 1.
A119996: Numerator of sum(k=1..n, 1 / ( Fibonacci(k) * Fibonacci(k+2) ) ).
A178314: Numbers n with property that n^2 contains "123" as a substring.
A236242: Numbers m with C(2*m, m) + prime(m) prime, where C(2*m, m) = (2*m)!/(m!)^2.
A238340: Number of partitions of 4n into 4 parts.
A251326: T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with every 2X2 subblock having one or two 1s

Sunday, November 20, 2016

Number of the day: 7167591

Benoit Mandelbrot was born on this day 92 years ago.

Properties of the number 7167591:

7167591 = 32 × 17 × 79 × 593 is the 6680306th composite number and is not squarefree.
7167591 has 4 distinct prime factors, 24 divisors, 27 antidivisors and 4432896 totatives.
7167591 has a triangular digit sum 36 in base 10.
7167591 is the difference of 2 nonnegative squares in 12 ways.
7167591 is the sum of 2 positive triangular numbers.
7167591 is the difference of 2 positive pentagonal numbers in 3 ways.
7167591 is not the sum of 3 positive squares.
71675912 = 33729842 + 63243452 = 28660412 + 65696402 = 12798002 + 70524092 = 44480162 + 56204552 is the sum of 2 positive squares in 4 ways.
71675912 is the sum of 3 positive squares.
7167591 is a divisor of 187348 - 1.
7167591 = '7167' + '591' is the concatenation of 2 semiprime numbers.
7167591 is palindromic in (at least) base 78.

Saturday, November 19, 2016

Number of the day: 7858960767033

Properties of the number 7858960767033:

7858960767033 = 3 × 6563 × 399154897 is a sphenic number and squarefree.
7858960767033 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 5238508855104 totatives.
7858960767033 has a semiprime digit sum 69 in base 10.
7858960767033 = 39294803835172 - 39294803835162 = 13098267945072 - 13098267945042 = 5987356272 - 5987290642 = 1995872932 - 1995676042 is the difference of 2 nonnegative squares in 4 ways.
7858960767033 is the difference of 2 positive pentagonal numbers in 2 ways.
7858960767033 = 43002 + 69072 + 28033722 is the sum of 3 positive squares.
78589607670332 = 52844423860082 + 58170381644552 is the sum of 2 positive squares in 1 way.
78589607670332 is the sum of 3 positive squares.
7858960767033 is a divisor of 11711299233347 - 1.
7858960767033 = '7858' + '960767033' is the concatenation of 2 emirpimes.

Friday, November 18, 2016

Number of the day: 44149457

Properties of the number 44149457:

44149457 = 11 × 227 × 17681 is a sphenic number and squarefree.
44149457 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 39956800 totatives.
44149457 has a semiprime digit sum 38 in base 10.
44149457 = 220747292 - 220747282 = 20067992 - 20067882 = 973592 - 971322 = 100892 - 75922 is the difference of 2 nonnegative squares in 4 ways.
44149457 is the difference of 2 positive pentagonal numbers in 4 ways.
44149457 = 832 + 2822 + 66382 is the sum of 3 positive squares.
441494572 = 230498072 + 376547602 is the sum of 2 positive squares in 1 way.
441494572 is the sum of 3 positive squares.
44149457 is a divisor of 7579605 - 1.
44149457 = '441494' + '57' is the concatenation of 2 semiprime numbers.

Thursday, November 17, 2016

Number of the day: 724004

August Ferdinand Möbius was born on this day 226 years ago.

Properties of the number 724004:

724004 = 22 × 181001 is the 665659th composite number and is not squarefree.
724004 has 2 distinct prime factors, 6 divisors, 17 antidivisors and 362000 totatives.
724004 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 724004 results in a semiprime.
724004 = 1810022 - 1810002 is the difference of 2 nonnegative squares in 1 way.
724004 is the difference of 2 positive pentagonal numbers in 1 way.
724004 = 702 + 8482 is the sum of 2 positive squares in 1 way.
724004 = 22 + 1082 + 8442 is the sum of 3 positive squares.
7240042 = 1187202 + 7142042 is the sum of 2 positive squares in 1 way.
7240042 is the sum of 3 positive squares.
724004 is a divisor of 1789125 - 1.

Wednesday, November 16, 2016

Number of the day: 79941605018971

Properties of the number 79941605018971:

79941605018971 = 421 × 189885047551 is semiprime and squarefree.
79941605018971 has 2 distinct prime factors, 4 divisors, 63 antidivisors and 79751719971000 totatives.
79941605018971 has a prime digit sum 67 in base 10.
Reversing the decimal digits of 79941605018971 results in a sphenic number.
79941605018971 = 399708025094862 - 399708025094852 = 949425239862 - 949425235652 is the difference of 2 nonnegative squares in 2 ways.
79941605018971 is the difference of 2 positive pentagonal numbers in 2 ways.
79941605018971 = 298912 + 507392 + 89408132 is the sum of 3 positive squares.
799416050189712 = 55066663789792 + 797517199714202 is the sum of 2 positive squares in 1 way.
799416050189712 is the sum of 3 positive squares.
79941605018971 is a divisor of 19496904910820 - 1.
79941605018971 = '799416050189' + '71' is the concatenation of 2 prime numbers.
79941605018971 is an emirpimes in (at least) the following bases: 18, 22, 24, 29, 35, 39, 41, 45, 49, 50, 54, 55, 57, 63, 69, 70, 71, 72, 73, 76, 87, 88, 91, 94, and 100.

Tuesday, November 15, 2016

Number of the day: 7230

Properties of the number 7230:

7230 = 2 × 3 × 5 × 241 is the 6305th composite number and is squarefree.
7230 has 4 distinct prime factors, 16 divisors, 9 antidivisors and 1920 totatives.
7230 has an oblong digit sum 12 in base 10.
7230 = 412 + … + 442 = 362 + … + 402 is the sum of at least 2 consecutive positive squares in 2 ways.
7230 is the sum of 2 positive triangular numbers.
7230 is the difference of 2 positive pentagonal numbers in 2 ways.
7230 = 12 + 22 + 852 is the sum of 3 positive squares.
72302 = 36002 + 62702 = 8822 + 71762 = 28562 + 66422 = 43382 + 57842 is the sum of 2 positive squares in 4 ways.
72302 is the sum of 3 positive squares.
7230 is a divisor of 6594 - 1.
7230 = '72' + '30' is the concatenation of 2 oblong numbers.
7230 is palindromic in (at least) the following bases: 22, 31, 52, and -29.
7230 in base 15 = 2220 and consists of only the digits '0' and '2'.
7230 in base 22 = eke and consists of only the digits 'e' and 'k'.
7230 in base 28 = 966 and consists of only the digits '6' and '9'.
7230 in base 31 = 7g7 and consists of only the digits '7' and 'g'.
7230 in base 42 = 446 and consists of only the digits '4' and '6'.
7230 in base 51 = 2dd and consists of only the digits '2' and 'd'.
7230 in base 52 = 2Z2 and consists of only the digits '2' and 'Z'.

The number 7230 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A027575: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2.
A027578: Sums of five consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2.
A031712: Least term in period of continued fraction for sqrt(n) is 34.
A059255: Both sum of n+1 consecutive squares and sum of the immediately following n consecutive squares.
A062681: Numbers that are sums of 2 or more consecutive squares in more than 1 way.
A062693: Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 3.
A110064: a(n+4) = a(n+1) - a(n), a(0) = 1, a(1) = -4, a(2) = 0, a(3) = 1.
A139275: a(n) = n*(8*n+1).
A229272: Numbers n for which n' + n and n' - n are both prime, n' being the arithmetic derivative of n.
A253805: a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypothenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804(n).

Monday, November 14, 2016

Number of the day: 670319

Properties of the number 670319:

670319 = 13 × 51563 is semiprime and squarefree.
670319 has 2 distinct prime factors, 4 divisors, 17 antidivisors and 618744 totatives.
670319 has an emirpimes digit sum 26 in base 10.
670319 = 3351602 - 3351592 = 257882 - 257752 is the difference of 2 nonnegative squares in 2 ways.
670319 is the difference of 2 positive pentagonal numbers in 2 ways.
670319 is not the sum of 3 positive squares.
6703192 = 2578152 + 6187562 is the sum of 2 positive squares in 1 way.
6703192 is the sum of 3 positive squares.
670319 is a divisor of 1307348 - 1.
670319 = '6703' + '19' is the concatenation of 2 prime numbers.
670319 = '6' + '70319' is the concatenation of 2 semiprime numbers.
670319 is an emirpimes in (at least) the following bases: 2, 4, 5, 6, 13, 14, 15, 16, 17, 21, 26, 28, 34, 35, 42, 44, 49, 54, 56, 59, 60, 63, 69, 70, 71, 72, 73, 75, 78, 81, 83, 86, 87, 89, 91, 92, 95, and 99.

Sunday, November 13, 2016

Number of the day: 8777

Properties of the number 8777:

8777 = 67 × 131 is semiprime and squarefree.
8777 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 8580 totatives.
8777 has a prime digit sum 29 in base 10.
Reversing the decimal digits of 8777 results in an emirpimes.
8777 = 43892 - 43882 = 992 - 322 is the difference of 2 nonnegative squares in 2 ways.
8777 is the difference of 2 positive pentagonal numbers in 1 way.
8777 = 122 + 132 + 922 is the sum of 3 positive squares.
87772 is the sum of 3 positive squares.
8777 is a divisor of 15716 - 1.
8777 = '877' + '7' is the concatenation of 2 prime numbers.
8777 = '87' + '77' is the concatenation of 2 semiprime numbers.
8777 is an emirpimes in (at least) the following bases: 3, 4, 6, 8, 10, 11, 12, 15, 17, 18, 19, 21, 22, 23, 27, 30, 31, 33, 34, 35, 37, 49, 50, 56, 57, 70, 71, 76, 78, 81, 82, 87, 88, 89, 90, 93, 94, 99, and 100.
8777 is palindromic in (at least) the following bases: 65, -9, -21, -32, -43, and -75.
8777 in base 8 = 21111 and consists of only the digits '1' and '2'.
8777 consists of only the digits '7' and '8'.
8777 in base 31 = 944 and consists of only the digits '4' and '9'.

The number 8777 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A033566: a(n) = (2*n+1) * (4*n-1).
A042753: Denominators of continued fraction convergents to sqrt(907).
A089372: Number of Motzkin paths of length n with no peaks at level 1.
A151793: Partial sums of A151782.
A153257: a(n)=n^3-(n+1)^2.
A178977: (3*n+2)*(3*n+5)/2.
A182988: The number of dominance pairs of integer partitions of n according to either/or dominance order, where dominance between two partitions x and y means that x is majorized by y or y is majorized by x.
A201362: Numbers n such that (2^n - n - 1)*2^n + 1 is prime.
A256340: Numbers which have only digits 7 and 8 in base 10.
A258349: Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)/2).

Saturday, November 12, 2016

Number of the day: 8889

Properties of the number 8889:

8889 = 3 × 2963 is semiprime and squarefree.
8889 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 5924 totatives.
8889 has a semiprime digit sum 33 in base 10.
8889 = (22 × 23)/2 + … + (39 × 40)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
8889 = 44452 - 44442 = 14832 - 14802 is the difference of 2 nonnegative squares in 2 ways.
8889 is the sum of 2 positive triangular numbers.
8889 is the difference of 2 positive pentagonal numbers in 1 way.
8889 = 22 + 72 + 942 is the sum of 3 positive squares.
88892 is the sum of 3 positive squares.
8889 is a divisor of 131481 - 1.
8889 is an emirpimes in (at least) the following bases: 9, 11, 12, 15, 18, 20, 23, 29, 34, 60, 63, 72, 77, 81, 84, 86, 87, 90, 95, and 99.
8889 is palindromic in (at least) the following bases: 25, 30, and 88.
8889 consists of only the digits '8' and '9'.
8889 in base 25 = e5e and consists of only the digits '5' and 'e'.
8889 in base 30 = 9q9 and consists of only the digits '9' and 'q'.

The number 8889 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A057534: a(n+1) = a(n)/2 if 2|a(n), a(n)/3 if 3|a(n), a(n)/5 if 5|a(n), a(n)/7 if 7|a(n), a(n)/11 if 11|a(n), a(n)/13 if 13|a(n), else 17*a(n)+1.
A059482: a(0)=1, a(n) = a(n-1) + 8*10^(n-1).
A080961: Fourth binomial transform of A010686 (period 2: repeat 1,5).
A125082: a(n) = n^4-n^3-n^2-n-1.
A178325: G.f.: A(x) = Sum_{n>=0} x^n/(1-x)^(n^2).
A178369: Numbers with rounded up arithmetic mean of digits = 9.
A213580: Principal diagonal of the convolution array A213579.
A220127: Number of tilings of a 10 X n rectangle using integer sided rectangular tiles of area 10.
A237346: For k in {2,3,...,9} define a sequence as follows: a(0)=0; for n>=0, a(n+1)=a(n)+1, unless a(n) ends in k, in which case a(n+1) is obtained by replacing the last digit of a(n) by the digit(s) of k^2. This is k(9).
A256341: Numbers which have only digits 8 and 9 in base 10.

Friday, November 11, 2016

Number of the day: 17411

Properties of the number 17411:

17411 = 23 × 757 is semiprime and squarefree.
17411 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 16632 totatives.
17411 has a semiprime digit sum 14 in base 10.
17411 has a triangular digit product 28 in base 10.
Reversing the decimal digits of 17411 results in a prime.
17411 = 87062 - 87052 = 3902 - 3672 is the difference of 2 nonnegative squares in 2 ways.
17411 is the difference of 2 positive pentagonal numbers in 2 ways.
17411 = 212 + 292 + 1272 is the sum of 3 positive squares.
174112 = 107642 + 136852 is the sum of 2 positive squares in 1 way.
174112 is the sum of 3 positive squares.
17411 is a divisor of 14274 - 1.
17411 is an emirpimes in (at least) the following bases: 7, 11, 14, 17, 19, 22, 30, 31, 34, 35, 36, 39, 43, 49, 52, 55, 56, 59, 67, 69, 71, 74, 76, 79, 86, 91, 94, 97, 98, and 99.
17411 is palindromic in (at least) the following bases: 3, 27, 68, -28, and -40.
17411 in base 3 = 212212212 and consists of only the digits '1' and '2'.
17411 in base 27 = nnn and consists of only the digit 'n'.
17411 in base 29 = kkb and consists of only the digits 'b' and 'k'.
17411 in base 39 = BHH and consists of only the digits 'B' and 'H'.

The number 17411 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A030534: Number of polyhexes of class PF2.
A037568: Base 3 digits are, in order, the first n terms of the periodic sequence with initial period 2,1,2.
A079968: Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={3}.
A177901: Numbers n > 1 such that log_10(n!) is closer to an integer than at any smaller n.
A250022: Numbers n > 1 such that the fractional part of log_10(n!) is greater than at any smaller n > 1.

Wednesday, November 9, 2016

Number of the day: 971671

Hermann Weyl was born on this day 131 years ago.

Properties of the number 971671:

971671 = 43 × 59 × 383 is a sphenic number and squarefree.
971671 has 3 distinct prime factors, 8 divisors, 17 antidivisors and 930552 totatives.
971671 has an emirp digit sum 31 in base 10.
Reversing the decimal digits of 971671 results in a prime.
971671 = 4858362 - 4858352 = 113202 - 112772 = 82642 - 82052 = 14602 - 10772 is the difference of 2 nonnegative squares in 4 ways.
971671 is the difference of 2 positive pentagonal numbers in 4 ways.
971671 is not the sum of 3 positive squares.
9716712 is the sum of 3 positive squares.
971671 is a divisor of 12971146 - 1.

Tuesday, November 8, 2016

Number of the day: 952547

Felix Hausdorff was born on this day 148 years ago.

Properties of the number 952547:

952547 is the 75094th prime.
952547 has 17 antidivisors and 952546 totatives.
Reversing the decimal digits of 952547 results in a semiprime.
952547 = 4762742 - 4762732 is the difference of 2 nonnegative squares in 1 way.
952547 is the difference of 2 positive pentagonal numbers in 1 way.
952547 = 432 + 632 + 9732 is the sum of 3 positive squares.
9525472 is the sum of 3 positive squares.
952547 is a divisor of 17325067 - 1.
952547 is an emirp in (at least) the following bases: 8, 11, 14, 15, 35, 37, 41, 43, 46, 49, 56, 59, 67, 69, 75, 80, 84, 85, 87, 88, 90, and 99.

Monday, November 7, 2016

Number of the day: 9935

Properties of the number 9935:

9935 = 5 × 1987 is semiprime and squarefree.
9935 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 7944 totatives.
9935 has an emirpimes digit sum 26 in base 10.
Reversing the decimal digits of 9935 results in a prime.
9935 = 49682 - 49672 = 9962 - 9912 is the difference of 2 nonnegative squares in 2 ways.
9935 is the difference of 2 positive pentagonal numbers in 2 ways.
9935 is not the sum of 3 positive squares.
99352 = 59612 + 79482 is the sum of 2 positive squares in 1 way.
99352 is the sum of 3 positive squares.
9935 is a divisor of 64712 - 1.
9935 is an emirpimes in (at least) the following bases: 2, 3, 5, 6, 11, 19, 21, 23, 24, 27, 30, 33, 35, 36, 37, 40, 44, 49, 59, 62, 67, 77, 79, 80, 84, 89, 92, and 100.
9935 is palindromic in (at least) the following bases: 52, and -77.
9935 in base 3 = 111121222 and consists of only the digits '1' and '2'.
9935 in base 31 = aaf and consists of only the digits 'a' and 'f'.
9935 in base 44 = 55Z and consists of only the digits '5' and 'Z'.
9935 in base 51 = 3ff and consists of only the digits '3' and 'f'.
9935 in base 52 = 3Z3 and consists of only the digits '3' and 'Z'.
9935 in base 57 = 33H and consists of only the digits '3' and 'H'.

The number 9935 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A147357: Number of n X n binary arrays symmetric about both diagonal and antidiagonal with all ones connected only in a 0110-0110-1111 pattern in any orientation.
A148616: 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, 0, 0), (-1, 0, 1), (0, -1, 0), (1, 0, 0), (1, 1, -1)}
A239852: Number of nX2 0..3 arrays with no element equal to zero plus the sum of elements to its left or zero plus the sum of elements above it or one plus the sum of the elements diagonally to its northwest or one plus the sum of the elements antidiagonally to its northeast, modulo 4

Sunday, November 6, 2016

Number of the day: 82801610348001

Properties of the number 82801610348001:

82801610348001 = 3 × 47 × 443 × 1325610527 is composite and squarefree.
82801610348001 has 4 distinct prime factors, 16 divisors, 43 antidivisors and 53904626429264 totatives.
82801610348001 has a sphenic digit sum 42 in base 10.
82801610348001 has an oblong digit sum 42 in base 10.
82801610348001 = 414008051740012 - 414008051740002 = 138002683913352 - 138002683913322 = 8808681952152 - 8808681951682 = 2936227318012 - 2936227316602 = 934555423752 - 934555419322 = 311518480492 - 311518467202 = 19884262012 - 19884053802 = 6628364952 - 6627740322 is the difference of 2 nonnegative squares in 8 ways.
82801610348001 is the sum of 2 positive triangular numbers.
82801610348001 is the difference of 2 positive pentagonal numbers in 4 ways.
82801610348001 = 17592 + 111642 + 90995322 is the sum of 3 positive squares.
828016103480012 is the sum of 3 positive squares.
82801610348001 is a divisor of 178722535378942 - 1.
82801610348001 = '828016103' + '48001' is the concatenation of 2 semiprime numbers.

Saturday, November 5, 2016

Number of the day: 460219

Properties of the number 460219:

460219 = 89 × 5171 is semiprime and squarefree.
460219 has 2 distinct prime factors, 4 divisors, 19 antidivisors and 454960 totatives.
460219 has a semiprime digit sum 22 in base 10.
460219 = 2301102 - 2301092 = 26302 - 25412 is the difference of 2 nonnegative squares in 2 ways.
460219 is the sum of 2 positive triangular numbers.
460219 is the difference of 2 positive pentagonal numbers in 2 ways.
460219 = 252 + 632 + 6752 is the sum of 3 positive squares.
4602192 = 2016692 + 4136802 is the sum of 2 positive squares in 1 way.
4602192 is the sum of 3 positive squares.
460219 is a divisor of 130322 - 1.
460219 is an emirpimes in (at least) the following bases: 4, 5, 8, 9, 11, 12, 13, 17, 19, 21, 23, 24, 25, 31, 34, 35, 43, 46, 47, 50, 56, 59, 60, 62, 64, 65, 67, 72, 74, 76, 78, 79, 81, 88, 92, 94, 96, 97, and 98.
460219 is palindromic in (at least) base -83.

Friday, November 4, 2016

Number of the day: 136868196

Properties of the number 136868196:

136868196 = 22 × 3 × 709 × 16087 is the 129123028th composite number and is not squarefree.
136868196 has 4 distinct prime factors, 24 divisors, 19 antidivisors and 45555552 totatives.
136868196 = 342170502 - 342170482 = 114056862 - 114056802 = 489702 - 475522 = 182142 - 139602 is the difference of 2 nonnegative squares in 4 ways.
136868196 is the sum of 2 positive triangular numbers.
136868196 is the difference of 2 positive pentagonal numbers in 2 ways.
136868196 = 462 + 4042 + 116922 is the sum of 3 positive squares.
1368681962 = 499983962 + 1274090402 is the sum of 2 positive squares in 1 way.
1368681962 is the sum of 3 positive squares.
136868196 is a divisor of 14092478 - 1.

Thursday, November 3, 2016

Number of the day: 20565182705512

Properties of the number 20565182705512:

20565182705512 = 23 × 17 × 109 × 307 × 4518859 is composite and not squarefree.
20565182705512 has 5 distinct prime factors, 64 divisors, 147 antidivisors and 9557710027776 totatives.
20565182705512 has an emirpimes digit sum 49 in base 10.
20565182705512 is the difference of 2 nonnegative squares in 16 ways.
20565182705512 is the sum of 2 positive triangular numbers.
20565182705512 is the difference of 2 positive pentagonal numbers in 8 ways.
20565182705512 = 9662 + 131202 + 45348662 is the sum of 3 positive squares.
205651827055122 = 98220111680402 + 180680612221122 = 96777330378882 + 181457494460402 = 19089106450882 + 204763961638802 = 113202840580802 + 171690974880882 is the sum of 2 positive squares in 4 ways.
205651827055122 is the sum of 3 positive squares.
20565182705512 is a divisor of 6139037716 - 1.
20565182705512 = '2056518270551' + '2' is the concatenation of 2 prime numbers.

Wednesday, November 2, 2016

Number of the day: 660647

George Boole was born on this day 201 years ago.

Properties of the number 660647:

660647 = 13 × 89 × 571 is a sphenic number and squarefree.
660647 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 601920 totatives.
660647 has a prime digit sum 29 in base 10.
Reversing the decimal digits of 660647 results in a sphenic number.
660647 = 3303242 - 3303232 = 254162 - 254032 = 37562 - 36672 = 8642 - 2932 is the difference of 2 nonnegative squares in 4 ways.
660647 is the sum of 2 positive triangular numbers.
660647 is the difference of 2 positive pentagonal numbers in 3 ways.
660647 is not the sum of 3 positive squares.
6606472 = 2894972 + 5938402 = 388282 + 6595052 = 4368152 + 4956282 = 2540952 + 6098282 is the sum of 2 positive squares in 4 ways.
6606472 is the sum of 3 positive squares.
660647 is a divisor of 109132 - 1.
660647 is palindromic in (at least) base 96.
660647 in base 34 = grgr and consists of only the digits 'g' and 'r'.

Tuesday, November 1, 2016

Number of the day: 5080

Properties of the number 5080:

5080 = 23 × 5 × 127 is the 4401th composite number and is not squarefree.
5080 has 3 distinct prime factors, 16 divisors, 7 antidivisors and 2016 totatives.
5080 has an emirp digit sum 13 in base 10.
5080 has a Fibonacci digit sum 13 in base 10.
5080 = 12712 - 12692 = 6372 - 6332 = 2592 - 2492 = 1372 - 1172 is the difference of 2 nonnegative squares in 4 ways.
5080 is the difference of 2 positive pentagonal numbers in 2 ways.
5080 = 182 + 202 + 662 is the sum of 3 positive squares.
50802 = 30482 + 40642 is the sum of 2 positive squares in 1 way.
50802 is the sum of 3 positive squares.
5080 is a divisor of 5092 - 1.
5080 is palindromic in (at least) the following bases: -35, and -36.
5080 in base 22 = aak and consists of only the digits 'a' and 'k'.
5080 in base 35 = 455 and consists of only the digits '4' and '5'.

The number 5080 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A086417: Sum of divisors of 3-smooth numbers.
A100155: Structured truncated octahedral numbers.
A104494: Positive integers n such that n^17 + 1 is semiprime (A001358).
A173723: Number of symmetry classes of 3x3 semimagic squares with distinct positive values < n.
A175926: Sum of divisors of cubes.
A177011: Define two triangular arrays by: B(0,0)=C(0,0)=1, B(0,r)=C(0,r)=0 for r>0, C(t,-1)=C(t,0); and for t,r >= 0, B(t+1,r)=C(t,r-1)+2C(t,r)-B(t,r), C(t+1,r)=B(t+1,r)+2B(t+1,r+1)-C(t,r). Sequence gives array B(t,r) read by rows.
A189449: T(n,k)=Number of nXk array permutations with each element moving zero or one space horizontally or diagonally
A239056: Sum of the parts in the partitions of 4n into 4 parts with smallest part = 1.
A256708: Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 8 as largest digit.
A264643: T(n,k)=Number of nXk arrays of permutations of 0..n*k-1 with rows nondecreasing modulo 3 and columns nondecreasing modulo 4.