Thursday, August 31, 2017

Number of the day: 963994

Properties of the number 963994:

963994 = 2 × 481997 is semiprime and squarefree.
963994 has 2 distinct prime factors, 4 divisors, 37 antidivisors and 481996 totatives.
Reversing the decimal digits of 963994 results in a sphenic number.
963994 is the sum of 2 positive triangular numbers.
963994 is the difference of 2 positive pentagonal numbers in 2 ways.
963994 = 6752 + 7132 is the sum of 2 positive squares in 1 way.
963994 = 932 + 1282 + 9692 is the sum of 3 positive squares.
9639942 = 527442 + 9625502 is the sum of 2 positive squares in 1 way.
9639942 is the sum of 3 positive squares.
963994 is a proper divisor of 232939 - 1.
963994 is an emirpimes in (at least) the following bases: 5, 19, 23, 24, 28, 29, 31, 32, 41, 43, 57, 73, 74, 75, 78, 87, 88, 90, 93, and 94.

Wednesday, August 30, 2017

Number of the day: 43292

Properties of the number 43292:

43292 = 22 × 79 × 137 is the 38773th composite number and is not squarefree.
43292 has 3 distinct prime factors, 12 divisors, 27 antidivisors and 21216 totatives.
43292 has an oblong digit sum 20 in base 10.
Reversing the decimal digits of 43292 results in a sphenic number.
43292 = 108242 - 108222 = 2162 - 582 is the difference of 2 nonnegative squares in 2 ways.
43292 is the difference of 2 positive pentagonal numbers in 2 ways.
43292 is not the sum of 3 positive squares.
432922 = 278082 + 331802 is the sum of 2 positive squares in 1 way.
432922 is the sum of 3 positive squares.
43292 is a proper divisor of 82326 - 1.
43292 is palindromic in (at least) the following bases: -63, and -67.
43292 in base 24 = 333k and consists of only the digits '3' and 'k'.
43292 in base 62 = BGG and consists of only the digits 'B' and 'G'.

Tuesday, August 29, 2017

Number of the day: 351

Properties of the number 351:

351 = 33 × 13 is the 280th composite number and is not squarefree.
351 has 2 distinct prime factors, 8 divisors, 9 antidivisors and 216 totatives.
351 has a semiprime digit sum 9 in base 10.
351 has an emirpimes digit product 15 in base 10.
351 has a triangular digit product 15 in base 10.
Reversing the decimal digits of 351 results in a triangular number.
351 = 73 + 23 is the sum of 2 positive cubes in 1 way.
351 = 1762 - 1752 = 602 - 572 = 242 - 152 = 202 - 72 is the difference of 2 nonnegative squares in 4 ways.
351 = (26 × 27)/2 is a triangular number.
351 is not the sum of 3 positive squares.
3512 = 1352 + 3242 is the sum of 2 positive squares in 1 way.
3512 is the sum of 3 positive squares.
351 is a proper divisor of 532 - 1.
351 = '35' + '1' is the concatenation of 2 pentagonal numbers.
351 is palindromic in (at least) the following bases: 14, 26, 38, -7, -12, -25, -35, -50, and -70.
351 in base 3 = 111000 and consists of only the digits '0' and '1'.
351 in base 4 = 11133 and consists of only the digits '1' and '3'.
351 in base 7 = 1011 and consists of only the digits '0' and '1'.
351 in base 14 = 1b1 and consists of only the digits '1' and 'b'.
351 in base 18 = 119 and consists of only the digits '1' and '9'.

The number 351 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000217: Triangular numbers: a(n) = binomial(n+1,2) = n(n+1)/2 = 0 + 1 + 2 + ... + n.
A000931: Padovan sequence (or Padovan numbers): a(n) = a(n-2) + a(n-3) with a(0)=1, a(1)=a(2)=0.
A003325: Numbers that are the sum of 2 positive cubes.
A005251: a(0) = 0, a(1) = a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).
A005836: Numbers n whose base 3 representation contains no 2.
A014105: Second hexagonal numbers: a(n) = n*(2n+1).
A014580: Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.
A014613: Numbers that are products of 4 primes (these numbers are sometimes called "4-almost primes", a generalization of semiprimes).
A074377: Generalized 10-gonal numbers: n*(4*n-3), n=0, +- 1, +- 2, +- 3,...
A084438: Positive integers n such that n!!!-1 = A007661(n)-1 is prime.

Monday, August 28, 2017

Number of the day: 5001

Properties of the number 5001:

5001 is a cyclic number.
5001 = 3 × 1667 is semiprime and squarefree.
5001 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 3332 totatives.
5001 has a semiprime digit sum 6 in base 10.
5001 has a triangular digit sum 6 in base 10.
5001 has an oblong digit sum 6 in base 10.
Reversing the decimal digits of 5001 results in a sphenic number.
5001 = (14 × 15)/2 + … + (31 × 32)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
5001 = 25012 - 25002 = 8352 - 8322 is the difference of 2 nonnegative squares in 2 ways.
5001 is the sum of 2 positive triangular numbers.
5001 is the difference of 2 positive pentagonal numbers in 1 way.
5001 = 12 + 102 + 702 is the sum of 3 positive squares.
50012 is the sum of 3 positive squares.
5001 is a proper divisor of 87717 - 1.
5001 is an emirpimes in (at least) the following bases: 7, 11, 14, 19, 25, 29, 30, 38, 40, 43, 45, 49, 51, 54, 55, 57, 60, 61, 66, 67, 68, 71, 75, 76, 78, 79, 83, 84, 87, 89, 90, 91, 93, 94, and 95.
5001 is palindromic in (at least) the following bases: 6, 8, 18, 27, -23, -24, -42, -49, and -100.
5001 in base 8 = 11611 and consists of only the digits '1' and '6'.
5001 in base 9 = 6766 and consists of only the digits '6' and '7'.
5001 in base 18 = f7f and consists of only the digits '7' and 'f'.
5001 in base 22 = a77 and consists of only the digits '7' and 'a'.
5001 in base 23 = 9aa and consists of only the digits '9' and 'a'.
5001 in base 27 = 6n6 and consists of only the digits '6' and 'n'.

The number 5001 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A033819: Trimorphic numbers: n^3 ends with n. Also m-morphic numbers for all m>5 such that m-1 is not divisible by 10 and m==3 (mod 4).
A069860: Numbers n that divide the concatenation of n+1 and n+2.
A081585: Third row of Pascal-(1,3,1) array A081578.
A105476: Number of compositions of n when each even part can be of two kinds.
A136811: Numbers n such that n and the square of n use only the digits 0, 1, 2, 3 and 5.
A144390: a(n) = 3*n^2 - n - 1.
A216899: Smallest palindromic number of 5 digits in two bases differing by n.
A255830: Numbers D such that D^2 = A^4 + B^5 + C^6 for some positive integers A, B, C.
A256631: Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 5 as largest digit.
A259384: Palindromic numbers in bases 6 and 8 written in base 10.

Sunday, August 27, 2017

Number of the day: 7215662483

Giuseppe Peano was born on this day 159 years ago.

Properties of the number 7215662483:

7215662483 is a cyclic number.
7215662483 = 131 × 55081393 is semiprime and squarefree.
7215662483 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 7160580960 totatives.
Reversing the decimal digits of 7215662483 results in a sphenic number.
7215662483 = 8153 + 10933 + 17513 is the sum of 3 positive cubes in 1 way.
7215662483 = 36078312422 - 36078312412 = 275407622 - 275406312 is the difference of 2 nonnegative squares in 2 ways.
7215662483 is the difference of 2 positive pentagonal numbers in 2 ways.
7215662483 = 72 + 972 + 849452 is the sum of 3 positive squares.
72156624832 = 31732162752 + 64804693922 is the sum of 2 positive squares in 1 way.
72156624832 is the sum of 3 positive squares.
7215662483 is a proper divisor of 5415158920 - 1.
7215662483 is an emirpimes in (at least) the following bases: 2, 3, 8, 11, 12, 13, 16, 17, 18, 19, 21, 24, 25, 29, 30, 33, 35, 43, 44, 47, 50, 52, 54, 57, 59, 60, 62, 65, 66, 69, 74, 82, 84, 90, 92, and 95.

Saturday, August 26, 2017

Number of the day: 4803

Properties of the number 4803:

4803 is a cyclic number.
4803 = 3 × 1601 is semiprime and squarefree.
4803 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 3200 totatives.
4803 has an emirpimes digit sum 15 in base 10.
4803 has a triangular digit sum 15 in base 10.
4803 = 24022 - 24012 = 8022 - 7992 is the difference of 2 nonnegative squares in 2 ways.
4803 is the sum of 2 positive triangular numbers.
4803 is the difference of 2 positive pentagonal numbers in 1 way.
4803 = 52 + 172 + 672 is the sum of 3 positive squares.
48032 = 2402 + 47972 is the sum of 2 positive squares in 1 way.
48032 is the sum of 3 positive squares.
4803 is a proper divisor of 1635 - 1.
4803 = '4' + '803' is the concatenation of 2 semiprime numbers.
4803 is an emirpimes in (at least) the following bases: 2, 15, 17, 21, 22, 23, 25, 30, 31, 33, 36, 37, 39, 56, 60, 63, 69, 73, 75, 76, 83, 84, 85, 86, 93, and 99.
4803 is palindromic in (at least) the following bases: 40, -18, -40, -48, and -98.
4803 in base 18 = eef and consists of only the digits 'e' and 'f'.
4803 in base 24 = 883 and consists of only the digits '3' and '8'.
4803 in base 39 = 366 and consists of only the digits '3' and '6'.
4803 in base 40 = 303 and consists of only the digits '0' and '3'.

The number 4803 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A050797: Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in exactly one way.
A095066: Number of fib001 primes (A095086) in range ]2^n,2^(n+1)].
A105632: Triangle, read by rows, where the g.f. A(x,y) satisfies the equation: A(x,y) = 1/(1-x*y) + x*A(x,y) + x^2*A(x,y)^2.
A120041: Number of 10-almost primes k such that 2^n < k <= 2^(n+1).
A160422: Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose virtual skeleton is a polyedge as the toothpick structure of A139250 but with toothpicks of length 6.
A182714: Number of 4's in the last section of the set of partitions of n.
A188536: Potential magic constants of 7 X 7 magic squares composed of consecutive primes.
A206920: Sum of the first n binary palindromes; a(n) = sum(k=1..n, A006995(k)).
A260052: Composites whose prime factorization in base 8 is an anagram of the number in base 8.
A261780: Number A(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Friday, August 25, 2017

Number of the day: 45489434

Properties of the number 45489434:

45489434 = 2 × 22744717 is semiprime and squarefree.
45489434 has 2 distinct prime factors, 4 divisors, 13 antidivisors and 22744716 totatives.
45489434 has a prime digit sum 41 in base 10.
45489434 has sum of divisors equal to 68234154 which is a sphenic number.
Reversing the decimal digits of 45489434 results in a sphenic number.
45489434 = 29752 + 60532 is the sum of 2 positive squares in 1 way.
45489434 = 1372 + 1722 + 67412 is the sum of 3 positive squares.
454894342 = 277881842 + 360153502 is the sum of 2 positive squares in 1 way.
454894342 is the sum of 3 positive squares.
45489434 is a proper divisor of 71895393 - 1.
45489434 = '4548943' + '4' is the concatenation of 2 semiprime numbers.
45489434 = '45489' + '434' is the concatenation of 2 sphenic numbers.
45489434 is an emirpimes in (at least) the following bases: 7, 9, 11, 23, 62, 66, 67, 70, 72, 74, 77, 80, 86, 91, 93, 96, and 99.

Thursday, August 24, 2017

Number of the day: 2594

Properties of the number 2594:

2594 = 2 × 1297 is semiprime and squarefree.
2594 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 1296 totatives.
2594 has an oblong digit sum 20 in base 10.
2594 = (34 × 35)/2 + … + (37 × 38)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
2594 is the sum of 2 positive triangular numbers.
2594 = 352 + 372 is the sum of 2 positive squares in 1 way.
2594 = 72 + 122 + 492 is the sum of 3 positive squares.
25942 = 1442 + 25902 is the sum of 2 positive squares in 1 way.
25942 is the sum of 3 positive squares.
2594 is a proper divisor of 16636 - 1.
2594 = '25' + '94' is the concatenation of 2 semiprime numbers.
2594 is an emirpimes in (at least) the following bases: 4, 8, 11, 16, 21, 23, 26, 28, 30, 35, 39, 41, 43, 45, 46, 48, 55, 61, 62, 63, 67, 68, 69, 71, 75, 76, 81, 82, 83, 86, 89, 96, and 98.
2594 is palindromic in (at least) the following bases: 6, 32, 36, -6, -36, and -48.
2594 in base 4 = 220202 and consists of only the digits '0' and '2'.
2594 in base 6 = 20002 and consists of only the digits '0' and '2'.
2594 in base 16 = a22 and consists of only the digits '2' and 'a'.
2594 in base 31 = 2ll and consists of only the digits '2' and 'l'.
2594 in base 32 = 2h2 and consists of only the digits '2' and 'h'.
2594 in base 35 = 244 and consists of only the digits '2' and '4'.
2594 in base 36 = 202 and consists of only the digits '0' and '2'.
2594 in base 50 = 11i and consists of only the digits '1' and 'i'.

The number 2594 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005893: Number of points on surface of tetrahedron: 2n^2 + 2 (coordination sequence for sodalite net) for n>0.
A051989: Discriminants of real quadratic fields with class number 2 and related continued fraction period length of 24.
A077068: Semiprimes of form prime + 1.
A078164: Numbers n such that phi(n) is a perfect biquadrate.
A092228: Numbers n such that numerator of Bernoulli(2n) is divisible by 233, the ninth irregular prime.
A126171: Number of infinitary amicable pairs (i,j) with i<j and i<=10^n.
A135110: Positive numbers such that the digital sum base 2 and the digital sum base 10 are in a ratio of 2:10.
A207815: Triangle of coefficients of Chebyshev's S(n,x-3) polynomials (exponents of x in increasing order).
A218897: T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, diagonal or antidiagonal neighbors in a random 0..1 nXk array
A231515: T(n,k)=Number of nXk 0..1 arrays with no element less than a strict majority of its horizontal, vertical, diagonal and antidiagonal neighbors

Wednesday, August 23, 2017

Number of the day: 73464

Properties of the number 73464:

73464 = 23 × 3 × 3061 is the 66209th composite number and is not squarefree.
73464 has 3 distinct prime factors, 16 divisors, 15 antidivisors and 24480 totatives.
73464 has a triangular digit product 2016 in base 10.
Reversing the decimal digits of 73464 results in a sphenic number.
73464 = 183672 - 183652 = 91852 - 91812 = 61252 - 61192 = 30672 - 30552 is the difference of 2 nonnegative squares in 4 ways.
73464 is the sum of 2 positive triangular numbers.
73464 = 22 + 522 + 2662 is the sum of 3 positive squares.
734642 = 158402 + 717362 is the sum of 2 positive squares in 1 way.
734642 is the sum of 3 positive squares.
73464 is a proper divisor of 186730 - 1.
73464 = '7346' + '4' is the concatenation of 2 semiprime numbers.
73464 is palindromic in (at least) base -61.
73464 in base 56 = NNm and consists of only the digits 'N' and 'm'.
73464 in base 60 = KOO and consists of only the digits 'K' and 'O'.

Tuesday, August 22, 2017

Number of the day: 3974

Properties of the number 3974:

3974 = 2 × 1987 is semiprime and squarefree.
3974 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 1986 totatives.
3974 has a prime digit sum 23 in base 10.
3974 has an oblong digit product 756 in base 10.
Reversing the decimal digits of 3974 results in a prime.
3974 = 302 + … + 332 is the sum of at least 2 consecutive positive squares in 1 way.
3974 = 12 + 22 + 632 is the sum of 3 positive squares.
39742 is the sum of 3 positive squares.
3974 is a proper divisor of 6473 - 1.
3974 = '39' + '74' is the concatenation of 2 semiprime numbers.
3974 is an emirpimes in (at least) the following bases: 5, 6, 8, 12, 13, 16, 19, 20, 25, 26, 28, 29, 35, 36, 38, 39, 42, 44, 47, 48, 54, 56, 57, 59, 60, 63, 64, 67, 68, 71, 74, 75, 80, 81, 84, 86, 91, and 100.
3974 is palindromic in (at least) the following bases: 17, and 27.
3974 in base 17 = dcd and consists of only the digits 'c' and 'd'.
3974 in base 26 = 5mm and consists of only the digits '5' and 'm'.
3974 in base 27 = 5c5 and consists of only the digits '5' and 'c'.
3974 in base 31 = 446 and consists of only the digits '4' and '6'.
3974 in base 44 = 22E and consists of only the digits '2' and 'E'.

The number 3974 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A007684: Prime(n)*...*prime(a(n)) is the least product of consecutive primes that is non-deficient.
A007707: Prime(n)*...*prime(a(n)) is the least product of consecutive primes which is abundant.
A027575: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2.
A059886: a(n) = |{m : multiplicative order of 4 mod m=n}|.
A068336: a(1) = 1; a(n+1) = 1 + sum{k|n} a(k), sum is over the positive divisors, k, of n.
A081268: Diagonal of triangular spiral in A051682.
A104577: Indices of prime generalized tetranacci numbers, A073817.
A191455: Dispersion of (floor(n*e)), by antidiagonals.
A218064: T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal or antidiagonal neighbors in a random 0..1 nXk array
A227743: Integers n for which A173318(n) is a square.

Monday, August 21, 2017

Number of the day: 918776

Augustin-Louis Cauchy was born on this day 228 years ago.

Properties of the number 918776:

918776 = 23 × 114847 is the 846129th composite number and is not squarefree.
918776 has 2 distinct prime factors, 8 divisors, 9 antidivisors and 459384 totatives.
918776 has a semiprime digit sum 38 in base 10.
Reversing the decimal digits of 918776 results in a semiprime.
918776 = 2296952 - 2296932 = 1148492 - 1148452 is the difference of 2 nonnegative squares in 2 ways.
918776 = 1162 + 1342 + 9422 is the sum of 3 positive squares.
9187762 is the sum of 3 positive squares.
918776 is a proper divisor of 7319141 - 1.
918776 = '91877' + '6' is the concatenation of 2 semiprime numbers.

Sunday, August 20, 2017

Number of the day: 43040741

Properties of the number 43040741:

43040741 is a cyclic number.
43040741 = 31 × 1388411 is semiprime and squarefree.
43040741 has 2 distinct prime factors, 4 divisors, 23 antidivisors and 41652300 totatives.
43040741 has a prime digit sum 23 in base 10.
Reversing the decimal digits of 43040741 results in an emirpimes.
43040741 = 215203712 - 215203702 = 6942212 - 6941902 is the difference of 2 nonnegative squares in 2 ways.
43040741 is the difference of 2 positive pentagonal numbers in 2 ways.
43040741 = 492 + 1762 + 65582 is the sum of 3 positive squares.
430407412 is the sum of 3 positive squares.
43040741 is a proper divisor of 557277682 - 1.
43040741 is an emirpimes in (at least) the following bases: 6, 8, 9, 10, 11, 13, 16, 19, 21, 23, 27, 28, 31, 37, 41, 42, 47, 49, 53, 55, 58, 62, 69, 70, 71, 73, 75, 77, 78, 81, 86, 89, 90, 93, and 98.

Saturday, August 19, 2017

Number of the day: 61618

Properties of the number 61618:

61618 = 2 × 30809 is semiprime and squarefree.
61618 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 30808 totatives.
61618 has a semiprime digit sum 22 in base 10.
61618 is the sum of 2 positive triangular numbers.
61618 is the difference of 2 positive pentagonal numbers in 2 ways.
61618 = 1372 + 2072 is the sum of 2 positive squares in 1 way.
61618 = 432 + 602 + 2372 is the sum of 3 positive squares.
616182 = 240802 + 567182 is the sum of 2 positive squares in 1 way.
616182 is the sum of 3 positive squares.
61618 is a proper divisor of 73851 - 1.
61618 = '6' + '1618' is the concatenation of 2 semiprime numbers.
61618 is an emirpimes in (at least) the following bases: 4, 6, 9, 11, 13, 16, 22, 23, 25, 32, 35, 37, 44, 54, 56, 59, 60, 68, 73, 74, 76, 77, 88, 89, 97, and 100.
61618 is palindromic in (at least) the following bases: 41, and 49.
61618 in base 7 = 344434 and consists of only the digits '3' and '4'.
61618 in base 41 = aQa and consists of only the digits 'Q' and 'a'.
61618 in base 49 = PWP and consists of only the digits 'P' and 'W'.
61618 in base 55 = KKI and consists of only the digits 'I' and 'K'.
61618 in base 58 = IIM and consists of only the digits 'I' and 'M'.

The number 61618 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A187858: Number of 3-step one space for components leftwards or up, two space for components rightwards or down asymmetric quasi-queen's tours (antidiagonal moves become knight moves) on an n X n board summed over all starting positions

Friday, August 18, 2017

Number of the day: 4603

Properties of the number 4603:

4603 is a cyclic number.
4603 is the 623th prime.
4603 has 21 antidivisors and 4602 totatives.
4603 has an emirp digit sum 13 in base 10.
4603 has a Fibonacci digit sum 13 in base 10.
4603 = 23022 - 23012 is the difference of 2 nonnegative squares in 1 way.
4603 is the sum of 2 positive triangular numbers.
4603 is the difference of 2 positive pentagonal numbers in 1 way.
4603 = 32 + 252 + 632 is the sum of 3 positive squares.
46032 is the sum of 3 positive squares.
4603 is a proper divisor of 1793 - 1.
4603 is an emirp in (at least) the following bases: 4, 13, 17, 19, 22, 24, 29, 33, 35, 37, 38, 49, 50, 51, 55, 65, 67, 71, 75, 77, 78, 81, 83, 86, 93, 95, and 96.
4603 is palindromic in (at least) the following bases: 43, 59, -40, -46, and -78.
4603 in base 17 = ffd and consists of only the digits 'd' and 'f'.
4603 in base 39 = 311 and consists of only the digits '1' and '3'.
4603 in base 42 = 2PP and consists of only the digits '2' and 'P'.
4603 in base 43 = 2L2 and consists of only the digits '2' and 'L'.
4603 in base 58 = 1LL and consists of only the digits '1' and 'L'.
4603 in base 59 = 1J1 and consists of only the digits '1' and 'J'.

The number 4603 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A031936: Lower prime of a difference of 18 between consecutive primes.
A056108: Fourth spoke of a hexagonal spiral.
A066540: The first of two consecutive primes with equal digital sums.
A095673: Primes p = p_(n+1) such that p_n + p_(n+2) = 2*p_(n+1) + 12.
A100201: Primes of the form 23n+3.
A104824: Primes from merging of 4 successive digits in decimal expansion of Pi.
A113743: Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 6 multiples of n-1, n-2, ..., 1.
A198164: Primes from merging of 4 successive digits in decimal expansion of sqrt(2).
A272160: Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.
A275183: T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-1,0) or (-1,1) and new values introduced in order 0..2.

Thursday, August 17, 2017

Number of the day: 412485

Pierre de Fermat was born on this day 416 years ago.

Properties of the number 412485:

412485 is a cyclic number.
412485 = 3 × 5 × 107 × 257 is the 377675th composite number and is squarefree.
412485 has 4 distinct prime factors, 16 divisors, 23 antidivisors and 217088 totatives.
Reversing the decimal digits of 412485 results in a sphenic number.
412485 = 2062432 - 2062422 = 687492 - 687462 = 412512 - 412462 = 137572 - 137422 = 19812 - 18742 = 9312 - 6742 = 8032 - 4822 = 6532 - 1182 is the difference of 2 nonnegative squares in 8 ways.
412485 is the difference of 2 positive pentagonal numbers in 3 ways.
412485 = 202 + 712 + 6382 is the sum of 3 positive squares.
4124852 = 2044772 + 3582362 = 513602 + 4092752 = 2866532 + 2966042 = 2474912 + 3299882 is the sum of 2 positive squares in 4 ways.
4124852 is the sum of 3 positive squares.
412485 is a proper divisor of 64316 - 1.
412485 is palindromic in (at least) base -92.

Wednesday, August 16, 2017

Number of the day: 229585

Arthur Cayley was born on this day 196 years ago.

Properties of the number 229585:

229585 is a cyclic number.
229585 = 5 × 17 × 37 × 73 is the 209187th composite number and is squarefree.
229585 has 4 distinct prime factors, 16 divisors, 25 antidivisors and 165888 totatives.
229585 has an emirp digit sum 31 in base 10.
229585 = 1147932 - 1147922 = 229612 - 229562 = 67612 - 67442 = 31212 - 30842 = 16092 - 15362 = 13932 - 13082 = 7132 - 5282 = 4972 - 1322 is the difference of 2 nonnegative squares in 8 ways.
229585 is the sum of 2 positive triangular numbers.
229585 is the difference of 2 positive pentagonal numbers in 7 ways.
229585 = 2972 + 3762 = 1592 + 4522 = 2362 + 4172 = 882 + 4712 = 3242 + 3532 = 1922 + 4392 = 1442 + 4572 = 122 + 4792 is the sum of 2 positive squares in 8 ways.
229585 = 32 + 702 + 4742 is the sum of 3 positive squares.
2295852 = 1565852 + 1679002 = 1080402 + 2025752 = 365002 + 2266652 = 1467752 + 1765402 = 815852 + 2146002 = 75752 + 2294602 = 1215402 + 1947752 = 518002 + 2236652 = 235402 + 2283752 = 1478522 + 1756392 = 1181932 + 1968242 = 828962 + 2140972 = 1067992 + 2032322 = 351132 + 2268842 = 403692 + 2260082 = 1437362 + 1790232 = 1227292 + 1940282 = 531672 + 2233442 = 1594322 + 1651992 = 972362 + 2079772 = 245282 + 2282712 = 1558572 + 1685762 = 927592 + 2100122 = 196332 + 2287442 = 1316162 + 1881132 = 634922 + 2206312 = 114962 + 2292972 = 1377512 + 1836682 = 1290642 + 1898732 = 707372 + 2184162 = 1180482 + 1969112 = 478042 + 2245532 = 276082 + 2279192 = 903212 + 2110722 = 169832 + 2289562 = 581912 + 2220882 = 1509602 + 1729752 = 1146652 + 1989002 = 867002 + 2125852 = 744602 + 2171752 is the sum of 2 positive squares in 40 ways.
2295852 is the sum of 3 positive squares.
229585 is a proper divisor of 15118 - 1.

The number 229585 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A218135: Norm of coefficients in the expansion of 1 / (1 - x - 2*I*x^2), where I^2=-1.
A255214: Number of partitions of n^2 into at most 10 square parts.

Tuesday, August 15, 2017

Number of the day: 719628

Happy Pythagorean Theorem Day!

Properties of the number 719628:

719628 = 22 × 3 × 7 × 13 × 659 is the 661629th composite number and is not squarefree.
719628 has 5 distinct prime factors, 48 divisors, 19 antidivisors and 189504 totatives.
719628 has a semiprime digit sum 33 in base 10.
719628 = 1799082 - 1799062 = 599722 - 599662 = 257082 - 256942 = 138522 - 138262 = 85882 - 85462 = 46522 - 45742 = 20682 - 18862 = 9322 - 3862 is the difference of 2 nonnegative squares in 8 ways.
719628 is the difference of 2 positive pentagonal numbers in 3 ways.
719628 = 1062 + 2142 + 8142 is the sum of 3 positive squares.
7196282 = 2767802 + 6642722 is the sum of 2 positive squares in 1 way.
7196282 is the sum of 3 positive squares.
719628 is a proper divisor of 131912 - 1.

The number 719628 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A241576: Third differences of A001521.
A244020: Number of new points at the n-th step of the following iteration, starting with four points in general position in the real projective plane: dualize the current pointset to a family of lines, take all intersections of those lines, repeat.

Monday, August 14, 2017

Number of the day: 749736

Properties of the number 749736:

749736 = 23 × 34 × 13 × 89 is the 689518th composite number and is not squarefree.
749736 has 4 distinct prime factors, 80 divisors, 21 antidivisors and 228096 totatives.
749736 has a triangular digit sum 36 in base 10.
749736 is the difference of 2 nonnegative squares in 20 ways.
749736 is the sum of 2 positive triangular numbers.
749736 is the difference of 2 positive pentagonal numbers in 1 way.
749736 = 5942 + 6302 = 3062 + 8102 is the sum of 2 positive squares in 2 ways.
749736 = 562 + 1942 + 8422 is the sum of 3 positive squares.
7497362 = 3285362 + 6739202 = 440642 + 7484402 = 4957202 + 5624642 = 2883602 + 6920642 is the sum of 2 positive squares in 4 ways.
7497362 is the sum of 3 positive squares.
749736 is a proper divisor of 17918 - 1.
749736 in base 34 = j2j2 and consists of only the digits '2' and 'j'.
749736 in base 35 = hh11 and consists of only the digits '1' and 'h'.

The number 749736 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A157377: a(n) = 531441*n - 313146.

Sunday, August 13, 2017

Number of the day: 4548893344

Properties of the number 4548893344:

4548893344 = 25 × 1889 × 75253 is the 4334179637th composite number and is not squarefree.
4548893344 has 3 distinct prime factors, 24 divisors, 15 antidivisors and 2273212416 totatives.
Reversing the decimal digits of 4548893344 results in a sphenic number.
4548893344 = 11372233372 - 11372233352 = 5686116702 - 5686116662 = 2843058382 - 2843058302 = 1421529252 - 1421529092 = 6039132 - 6001352 = 3047902 - 2972342 = 1580622 - 1429502 = 903652 - 601412 is the difference of 2 nonnegative squares in 8 ways.
4548893344 is the sum of 2 positive triangular numbers.
4548893344 is the difference of 2 positive pentagonal numbers in 2 ways.
4548893344 = 419802 + 527882 = 64122 + 671402 is the sum of 2 positive squares in 2 ways.
4548893344 = 9602 + 28122 + 673802 is the sum of 3 positive squares.
45488933442 = 31570138562 + 32750105602 = 10242525442 + 44320804802 = 8610033602 + 44666658562 = 24800605442 + 38133620802 is the sum of 2 positive squares in 4 ways.
45488933442 is the sum of 3 positive squares.
4548893344 is a proper divisor of 33150168 - 1.

Saturday, August 12, 2017

Number of the day: 58483

Properties of the number 58483:

58483 is a cyclic number.
58483 = 233 × 251 is semiprime and squarefree.
58483 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 58000 totatives.
58483 has a triangular digit sum 28 in base 10.
Reversing the decimal digits of 58483 results in a sphenic number.
58483 = 292422 - 292412 = 2422 - 92 is the difference of 2 nonnegative squares in 2 ways.
58483 is the difference of 2 positive pentagonal numbers in 1 way.
58483 = 172 + 872 + 2252 is the sum of 3 positive squares.
584832 = 263552 + 522082 is the sum of 2 positive squares in 1 way.
584832 is the sum of 3 positive squares.
58483 is a proper divisor of 50329 - 1.
58483 is an emirpimes in (at least) the following bases: 5, 6, 8, 11, 14, 15, 16, 17, 20, 22, 25, 26, 27, 29, 33, 34, 38, 39, 41, 43, 46, 48, 50, 51, 54, 55, 56, 58, 59, 61, 64, 70, 72, 75, 76, 77, 81, 86, 87, 91, 92, and 99.
58483 is palindromic in (at least) the following bases: 45, 73, and -56.
58483 in base 45 = SdS and consists of only the digits 'S' and 'd'.
58483 in base 55 = JII and consists of only the digits 'I' and 'J'.

The number 58483 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A058518: Number of forests of B-trees of order 3 with n labeled leaves.
A069969: Numbers n such that phi(reverse(n)) = phi(reverse(n-1)) + phi(reverse(n-2)).
A249109: Composite numbers whose sum of aliquot parts divides the sum of the aliquot parts of the numbers less than or equal to n and not relatively prime to n.
A275251: Sequence of pairwise relatively prime numbers of class P_6 (see comment in A275246).

Friday, August 11, 2017

Number of the day: 717853

Properties of the number 717853:

717853 = 233 × 59 is the 659972th composite number and is not squarefree.
717853 has 2 distinct prime factors, 8 divisors, 31 antidivisors and 675004 totatives.
717853 has an emirp digit sum 31 in base 10.
Reversing the decimal digits of 717853 results in a semiprime.
717853 = 3589272 - 3589262 = 156172 - 155942 = 61132 - 60542 = 9432 - 4142 is the difference of 2 nonnegative squares in 4 ways.
717853 is the difference of 2 positive pentagonal numbers in 4 ways.
717853 = 292 + 362 + 8462 is the sum of 3 positive squares.
7178532 is the sum of 3 positive squares.
717853 is a proper divisor of 8271058 - 1.
717853 = '71' + '7853' is the concatenation of 2 prime numbers.

The number 717853 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A229146: n^3*(5*n+3)/2.

Thursday, August 10, 2017

Number of the day: 971

Properties of the number 971:

971 is a cyclic number.
971 is the 164th prime.
971 has 5 antidivisors and 970 totatives.
971 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 971 results in an emirp.
971 = 4862 - 4852 is the difference of 2 nonnegative squares in 1 way.
971 is the difference of 2 positive pentagonal numbers in 1 way.
971 = 12 + 32 + 312 is the sum of 3 positive squares.
9712 is the sum of 3 positive squares.
971 is a proper divisor of 23910 - 1.
971 is an emirp in (at least) the following bases: 3, 4, 7, 8, 10, 15, 16, 17, 23, 25, 33, 35, 37, 38, 41, 42, 43, 46, 49, 50, 53, 54, 55, 57, 59, 62, 64, 69, 70, 73, 77, 81, 82, 86, 87, 89, 93, 97, 98, and 100.
971 is palindromic in (at least) the following bases: 19, -14, and -97.
971 in base 7 = 2555 and consists of only the digits '2' and '5'.
971 in base 13 = 599 and consists of only the digits '5' and '9'.
971 in base 15 = 44b and consists of only the digits '4' and 'b'.
971 in base 18 = 2hh and consists of only the digits '2' and 'h'.
971 in base 19 = 2d2 and consists of only the digits '2' and 'd'.

The number 971 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000928: Irregular primes: p is regular if and only if the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) are not divisible by p.
A001913: Full reptend primes: primes with primitive root 10.
A005846: Primes of the form n^2 + n + 41.
A006567: Emirps (primes whose reversal is a different prime).
A007500: Primes whose reversal in base 10 is also prime (called "palindromic primes" by D. Wells, although that name usually refers to A002385). Also called reversible primes.
A030430: Primes of the form 10*n+1.
A046132: Larger member p+4 of cousin primes (p, p+4).
A061955: Numbers n such that n divides the (left) concatenation of all numbers <= n written in base 2 (most significant digit on right).
A141111: Primes of the form 4*x^2+x*y-4*y^2 (as well as of the form 4*x^2+9*x*y+y^2).
A235266: Primes whose base 2 representation is also the base 3 representation of a prime.

Wednesday, August 9, 2017

Number of the day: 3505

Properties of the number 3505:

3505 = 5 × 701 is semiprime and squarefree.
3505 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 2800 totatives.
3505 has an emirp digit sum 13 in base 10.
3505 has a Fibonacci digit sum 13 in base 10.
Reversing the decimal digits of 3505 results in an emirpimes.
3505 = 142 + … + 232 is the sum of at least 2 consecutive positive squares in 1 way.
3505 = 17532 - 17522 = 3532 - 3482 is the difference of 2 nonnegative squares in 2 ways.
3505 is the difference of 2 positive pentagonal numbers in 2 ways.
3505 = 362 + 472 = 162 + 572 is the sum of 2 positive squares in 2 ways.
3505 = 242 + 252 + 482 is the sum of 3 positive squares.
35052 = 21032 + 28042 = 18242 + 29932 = 9132 + 33842 = 13002 + 32552 is the sum of 2 positive squares in 4 ways.
35052 is the sum of 3 positive squares.
3505 is a proper divisor of 9115 - 1.
3505 is an emirpimes in (at least) the following bases: 4, 7, 10, 12, 13, 15, 16, 17, 18, 23, 26, 27, 29, 31, 37, 38, 39, 42, 45, 50, 51, 57, 61, 62, 65, 66, 68, 69, 71, 73, 74, 75, 79, 84, 87, 88, 89, 91, 92, 93, and 100.
3505 is palindromic in (at least) the following bases: 19, 22, 25, 34, 48, -8, -28, and -73.
3505 in base 8 = 6661 and consists of only the digits '1' and '6'.
3505 in base 19 = 9d9 and consists of only the digits '9' and 'd'.
3505 in base 21 = 7jj and consists of only the digits '7' and 'j'.
3505 in base 22 = 757 and consists of only the digits '5' and '7'.
3505 in base 25 = 5f5 and consists of only the digits '5' and 'f'.
3505 in base 29 = 44p and consists of only the digits '4' and 'p'.
3505 in base 33 = 377 and consists of only the digits '3' and '7'.
3505 in base 34 = 313 and consists of only the digits '1' and '3'.
3505 in base 47 = 1RR and consists of only the digits '1' and 'R'.
3505 in base 48 = 1P1 and consists of only the digits '1' and 'P'.

The number 3505 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A006367: Number of binary vectors of length n+1 beginning with 0 and containing just 1 singleton.
A013979: Expansion of 1/(1 - x^2 - x^3 - x^4) = 1/((1 + x)*(1 - x - x^3)).
A026905: Partial sums of the partition numbers A000041.
A072573: Odd interprimes not divisible by 3.
A098237: Composite de Polignac numbers (A006285).
A107458: G.f.: (1-x^2-x^3)/( (1+x)(1-x-x^3) ).
A136392: 6n^2 - 10n + 5.
A182840: Toothpick sequence on hexagonal net.
A205477: L.g.f.: Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + n*x^d/d).
A248112: Number T(n,k) of subsets of {1,...,n} containing n and having at least one set partition into k blocks with equal element sum; triangle T(n,k), n>=1, 1<=k<=floor((n+1)/2), read by rows.

Tuesday, August 8, 2017

Number of the day: 82545

Properties of the number 82545:

82545 = 3 × 5 × 5503 is a sphenic number and squarefree.
82545 has 3 distinct prime factors, 8 divisors, 9 antidivisors and 44016 totatives.
82545 = 412732 - 412722 = 137592 - 137562 = 82572 - 82522 = 27592 - 27442 is the difference of 2 nonnegative squares in 4 ways.
82545 is the difference of 2 positive pentagonal numbers in 2 ways.
82545 = 172 + 402 + 2842 is the sum of 3 positive squares.
825452 = 495272 + 660362 is the sum of 2 positive squares in 1 way.
825452 is the sum of 3 positive squares.
82545 is a proper divisor of 9296 - 1.
82545 = '82' + '545' is the concatenation of 2 semiprime numbers.
82545 in base 14 = 22121 and consists of only the digits '1' and '2'.

The number 82545 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A255550: Main diagonal of array A255551.

Monday, August 7, 2017

Number of the day: 45171

Properties of the number 45171:

45171 = 33 × 7 × 239 is the 40482th composite number and is not squarefree.
45171 has 3 distinct prime factors, 16 divisors, 23 antidivisors and 25704 totatives.
45171 = 225862 - 225852 = 75302 - 75272 = 32302 - 32232 = 25142 - 25052 = 10862 - 10652 = 8502 - 8232 = 3902 - 3272 = 2142 - 252 is the difference of 2 nonnegative squares in 8 ways.
45171 is the sum of 2 positive triangular numbers.
45171 is the difference of 2 positive pentagonal numbers in 1 way.
45171 = 112 + 792 + 1972 is the sum of 3 positive squares.
451712 is the sum of 3 positive squares.
45171 is a proper divisor of 75717 - 1.
45171 = '45' + '171' is the concatenation of 2 triangular numbers.
45171 in base 42 = PPL and consists of only the digits 'L' and 'P'.

The number 45171 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A031795: Period of continued fraction for sqrt(n) contains exactly 27 ones.
A050833: Numbers n such that 163*2^n-1 is a prime.
A095053: Number of primes with number of 1-bits equal to one plus number 0-bits (A095073) in range ]2^2n,2^(2n+1)].
A151350: Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (0, 1), (1, -1), (1, 1)}
A188381: Negabinary Keith numbers.
A241639: Number of partitions p of n such that (number of even numbers in p) >= (number of odd numbers in p).

Sunday, August 6, 2017

Number of the day: 995361

Johann Bernoulli was born on this day 350 years ago.

Properties of the number 995361:

995361 = 3 × 257 × 1291 is a sphenic number and squarefree.
995361 has 3 distinct prime factors, 8 divisors, 17 antidivisors and 660480 totatives.
995361 has a semiprime digit sum 33 in base 10.
Reversing the decimal digits of 995361 results in a sphenic number.
995361 = 4976812 - 4976802 = 1658952 - 1658922 = 20652 - 18082 = 10312 - 2602 is the difference of 2 nonnegative squares in 4 ways.
995361 is the sum of 2 positive triangular numbers.
995361 is the difference of 2 positive pentagonal numbers in 2 ways.
995361 = 802 + 1692 + 9802 is the sum of 3 positive squares.
9953612 = 1239362 + 9876152 is the sum of 2 positive squares in 1 way.
9953612 is the sum of 3 positive squares.
995361 is a proper divisor of 154315 - 1.
995361 = '995' + '361' is the concatenation of 2 semiprime numbers.

Saturday, August 5, 2017

Number of the day: 677201

Niels Henrik Abel was born on this day 215 years ago.

Properties of the number 677201:

677201 is a cyclic number.
677201 = 7 × 89 × 1087 is a sphenic number and squarefree.
677201 has 3 distinct prime factors, 8 divisors, 27 antidivisors and 573408 totatives.
677201 has a prime digit sum 23 in base 10.
677201 = 3386012 - 3386002 = 483752 - 483682 = 38492 - 37602 = 8552 - 2322 is the difference of 2 nonnegative squares in 4 ways.
677201 is the difference of 2 positive pentagonal numbers in 4 ways.
677201 = 242 + 652 + 8202 is the sum of 3 positive squares.
6772012 = 2967512 + 6087202 is the sum of 2 positive squares in 1 way.
6772012 is the sum of 3 positive squares.
677201 is a proper divisor of 257132 - 1.
677201 is palindromic in (at least) base -40.

Friday, August 4, 2017

Number of the day: 3119

William Rowan Hamilton was born on this day 212 years ago.

John Venn was born on this day 183 years ago.

Properties of the number 3119:

3119 is a cyclic number.
3119 and 3121 form a twin prime pair.
3119 has 21 antidivisors and 3118 totatives.
3119 has a semiprime digit sum 14 in base 10.
Reversing the decimal digits of 3119 results in a semiprime.
3119 = 15602 - 15592 is the difference of 2 nonnegative squares in 1 way.
3119 is the difference of 2 positive pentagonal numbers in 1 way.
3119 is not the sum of 3 positive squares.
31192 is the sum of 3 positive squares.
3119 is a proper divisor of 21559 - 1.
3119 = '31' + '19' is the concatenation of 2 prime numbers.
3119 is an emirp in (at least) the following bases: 2, 5, 15, 17, 19, 20, 24, 25, 27, 39, 42, 44, 46, 47, 51, 56, 58, 59, 68, 71, 72, 73, 76, 79, 81, 83, 85, 87, 88, 91, 92, 95, 97, and 100.
3119 is palindromic in (at least) the following bases: -15, and -38.
3119 in base 5 = 44434 and consists of only the digits '3' and '4'.
3119 in base 55 = 11d and consists of only the digits '1' and 'd'.

The number 3119 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001126: Primes with 7 as smallest primitive root.
A001154: Describe the previous term! (method A - initial term is 9).
A024770: Right-truncatable primes: every prefix is prime.
A045709: Primes with first digit 3.
A059452: Safe primes (A005385) which are not Sophie Germain primes.
A060229: Smaller member of a twin prime pair whose mean is a multiple of A002110(3)=30.
A068652: Numbers such that every cyclic permutation is a prime.
A142463: a(n) = 2*n^2 + 2*n - 1.
A164288: Members of A164368 which are not Ramanujan primes.
A215419: Primes that remain prime when a single digit 3 is inserted between any two consecutive digits or as the leading or trailing digit.

Thursday, August 3, 2017

Number of the day: 960

Properties of the number 960:

960 = 26 × 3 × 5 is the 797th composite number and is not squarefree.
960 has 3 distinct prime factors, 28 divisors, 7 antidivisors and 256 totatives.
960 has an emirpimes digit sum 15 in base 10.
960 has a triangular digit sum 15 in base 10.
960 = 62 + … + 142 is the sum of at least 2 consecutive positive squares in 1 way.
960 is the difference of 2 nonnegative squares in 10 ways.
960 is the difference of 2 positive pentagonal numbers in 1 way.
960 is not the sum of 3 positive squares.
9602 = 5762 + 7682 is the sum of 2 positive squares in 1 way.
9602 is the sum of 3 positive squares.
960 is a proper divisor of 312 - 1.
960 is palindromic in (at least) the following bases: 31, 39, 47, 59, 63, 79, and 95.
960 in base 4 = 33000 and consists of only the digits '0' and '3'.
960 in base 15 = 440 and consists of only the digits '0' and '4'.

The number 960 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000118: Number of ways of writing n as a sum of 4 squares; also theta series of lattice Z^4.
A002093: Highly abundant numbers: numbers n such that sigma(n) > sigma(m) for all m < n.
A002417: 4-dimensional figurate numbers: a(n) = n*binomial(n+2, 3).
A005179: Smallest number with exactly n divisors.
A005563: a(n) = n*(n+2) (or, (n+1)^2 - 1).
A025487: List giving least integer of each prime signature; also products of primorial numbers A002110.
A029470: Numbers n such that n divides the (right) concatenation of all numbers <= n written in base 25 (most significant digit on left).
A033996: 8 times triangular numbers: a(n) = 4n(n+1).
A051682: 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.
A075928: List of codewords in binary lexicode with Hamming distance 4 written as decimal numbers.

Tuesday, August 1, 2017

Number of the day: 73457

Properties of the number 73457:

73457 is a cyclic number.
73457 = 17 × 29 × 149 is a sphenic number and squarefree.
73457 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 66304 totatives.
73457 has an emirpimes digit sum 26 in base 10.
Reversing the decimal digits of 73457 results in a prime.
73457 = (48 × 49)/2 + … + (81 × 82)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
73457 = 367292 - 367282 = 21692 - 21522 = 12812 - 12522 = 3212 - 1722 is the difference of 2 nonnegative squares in 4 ways.
73457 is the difference of 2 positive pentagonal numbers in 4 ways.
73457 = 892 + 2562 = 42 + 2712 = 1842 + 1992 = 1242 + 2412 is the sum of 2 positive squares in 4 ways.
73457 = 142 + 192 + 2702 is the sum of 3 positive squares.
734572 = 345682 + 648152 = 102952 + 727322 = 490682 + 546652 = 230952 + 697322 = 21682 + 734252 = 455682 + 576152 = 196682 + 707752 = 57452 + 732322 = 427052 + 597682 = 506602 + 531932 = 293932 + 673202 = 326402 + 658072 = 251432 + 690202 is the sum of 2 positive squares in 13 ways.
734572 is the sum of 3 positive squares.
73457 is a proper divisor of 70128 - 1.
73457 = '7' + '3457' is the concatenation of 2 prime numbers.
73457 = '734' + '57' is the concatenation of 2 semiprime numbers.
73457 is palindromic in (at least) the following bases: 2, and 42.
73457 in base 42 = fQf and consists of only the digits 'Q' and 'f'.
73457 in base 56 = NNf and consists of only the digits 'N' and 'f'.

The number 73457 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A090508: Least number beginning with prime(n) such that every concatenation is a prime.
A253131: Number of length 3+2 0..n arrays with the sum of medians of adjacent triples multiplied by some arrangement of +-1 equal to zero