Thursday, April 26, 2018

Number of the day: 73037

Properties of the number 73037:

73037 is a cyclic number.
73037 and 73039 form a twin prime pair.
73037 has 19 antidivisors and 73036 totatives.
73037 has an oblong digit sum 20 in base 10.
73037 = 193 + 193 + 393 is the sum of 3 positive cubes in 1 way.
73037 = 365192 - 365182 is the difference of 2 nonnegative squares in 1 way.
73037 is the difference of 2 positive pentagonal numbers in 1 way.
73037 = 262 + 2692 is the sum of 2 positive squares in 1 way.
73037 = 42 + 112 + 2702 is the sum of 3 positive squares.
730372 = 139882 + 716852 is the sum of 2 positive squares in 1 way.
730372 is the sum of 3 positive squares.
73037 is a proper divisor of 647589 - 1.
73037 = '7' + '3037' is the concatenation of 2 prime numbers.
73037 is an emirp in (at least) the following bases: 4, 5, 9, 11, 13, 14, 20, 23, 28, 34, 36, 39, 40, 44, 47, 49, 54, 56, 57, 58, 61, 62, 71, 73, 74, 80, 89, 93, 96, 99, and 100.
73037 is a palindrome (in base 10).
73037 is palindromic in (at least) the following bases: 42, -46, and -61.
73037 in base 42 = fGf and consists of only the digits 'G' and 'f'.
73037 in base 60 = KHH and consists of only the digits 'H' and 'K'.

The number 73037 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A052024: Every suffix of palindromic prime a(n) is prime (left-truncatable).
A062352: Palindromic primes with strictly decreasing digits up to the middle and then strictly increasing.
A081220: Palindromic primes = 1 mod 4.
A082769: a(n) = smallest palindromic prime that begins with A082768(n), or 0 if no such number exists.
A083840: Palindromic primes which are a member of a twin prime pair.
A083841: Palindromic primes p such that p+2 is also a prime: members of A083840 which are the smaller member of a twin prime pair.
A088269: Palindromic primes that yield a prime when sandwiched between two 1's. (Prefixing and suffixing a one on both sides yields another pal prime).
A109184: Palindromic primes with digit sum 20.
A136098: Prime palindromic cyclops numbers.
A260378: Primes that contain only the digits (0, 3, 7).

Wednesday, April 25, 2018

Number of the day: 8442

Felix Klein was born on this day 169 years ago.

Andrey Kolmogorov was born on this day 115 years ago.

Properties of the number 8442:

8442 = 2 × 32 × 7 × 67 is the 7386th composite number and is not squarefree.
8442 has 4 distinct prime factors, 24 divisors, 17 antidivisors and 2376 totatives.
8442 is the sum of 2 positive triangular numbers.
8442 = 112 + 202 + 892 is the sum of 3 positive squares.
84422 is the sum of 3 positive squares.
8442 is a proper divisor of 9372 - 1.
8442 is palindromic in (at least) the following bases: 5, 37, -35, and -38.
8442 in base 5 = 232232 and consists of only the digits '2' and '3'.
8442 in base 20 = 1122 and consists of only the digits '1' and '2'.
8442 in base 26 = cci and consists of only the digits 'c' and 'i'.
8442 in base 34 = 7aa and consists of only the digits '7' and 'a'.
8442 in base 36 = 6ii and consists of only the digits '6' and 'i'.
8442 in base 37 = 666 and consists of only the digit '6'.

The number 8442 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001402: Number of partitions of n into at most 6 parts.
A022271: a(n) = n*(13*n + 1)/2.
A061428: Geometric mean of the digits = 4. In other words the product of the digits is = 4^k where k is the number of digits.
A129526: Number of n-bead two-color bracelets with 00 prohibited.
A135192: Numbers n that raised to the powers from 1 to k (with k>=1) are multiple of the sum of their digits (n raised to k+1 must not be a multiple). Case k=7.
A143860: Ulam's spiral (NNW spoke).
A161407: Number of partitions of n^2 into parts smaller than n.
A172179: (1,[99n+1]) Pascal Triangle.
A235497: 2n concatenated with n.
A299287: Coordination sequence for "tcd" 3D uniform tiling.

Tuesday, April 24, 2018

Number of the day: 42561

Properties of the number 42561:

42561 = 32 × 4729 is the 38110th composite number and is not squarefree.
42561 has 2 distinct prime factors, 6 divisors, 7 antidivisors and 28368 totatives.
42561 has an oblong digit product 240 in base 10.
42561 = 173 + 223 + 303 is the sum of 3 positive cubes in 1 way.
42561 = 212812 - 212802 = 70952 - 70922 = 23692 - 23602 is the difference of 2 nonnegative squares in 3 ways.
42561 is the difference of 2 positive pentagonal numbers in 1 way.
42561 = 1352 + 1562 is the sum of 2 positive squares in 1 way.
42561 = 342 + 912 + 1822 is the sum of 3 positive squares.
425612 = 61112 + 421202 is the sum of 2 positive squares in 1 way.
425612 is the sum of 3 positive squares.
42561 is a proper divisor of 12238 - 1.
42561 = '4' + '2561' is the concatenation of 2 semiprime numbers.
42561 = '42' + '561' is the concatenation of 2 sphenic numbers.
42561 is palindromic in (at least) base -72.

The number 42561 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005225: Number of permutations of length n with equal cycles.
A132960: a(n) = n!*Sum_{d|n} (-1)^(d+1)/(d!*(n/d)^d).
A155627: 6^n-4^n+1.
A218868: Triangular array read by rows: T(n,k) is the number of n-permutations that have exactly k distinct cycle lengths.

Monday, April 23, 2018

Number of the day: 6426

Properties of the number 6426:

6426 = 2 × 33 × 7 × 17 is the 5590th composite number and is not squarefree.
6426 has 4 distinct prime factors, 32 divisors, 17 antidivisors and 1728 totatives.
6426 is the sum of 2 positive triangular numbers.
6426 is the difference of 2 positive pentagonal numbers in 1 way.
6426 = 162 + 292 + 732 is the sum of 3 positive squares.
64262 = 30242 + 56702 is the sum of 2 positive squares in 1 way.
64262 is the sum of 3 positive squares.
6426 is a proper divisor of 6133 - 1.
6426 is palindromic in (at least) the following bases: 38, and -73.
6426 in base 37 = 4PP and consists of only the digits '4' and 'P'.
6426 in base 38 = 4H4 and consists of only the digits '4' and 'H'.
6426 in base 56 = 22g and consists of only the digits '2' and 'g'.

The number 6426 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A014640: Even heptagonal numbers (A000566).
A033571: a(n) = (2*n + 1)*(5*n + 1).
A050047: a(n) = a(n-1)+a(m), where m=2n-2-2^(p+1) and 2^p<n-1<=2^(p+1), for n >= 4.
A073268: Number of plane binary trees whose right (or respectively: left) subtree is a unique "complete" tree of (2^m)-1 nodes with all the leaf-nodes at the same depth m and the left (or respectively: right) subtree is any plane binary tree of size n - 2^m + 1.
A078868: Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).
A171256: Numbers n such that sigma(n) = 10*phi(n) (where sigma=A000203, phi=A000010).
A179670: Numbers of the form p^3*q*r*s where p, q, r, and s are distinct primes.
A241647: Numbers m such that the GCD of the x's that satisfy sigma(x)=m is 2.
A277665: 5th-order coefficients of the 1/N-expansion of traces of negative powers of real Wishart matrices with parameter c=2.
A286416: Number T(n,k) of entries in the k-th last blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Sunday, April 22, 2018

Number of the day: 8088

Properties of the number 8088:

8088 = 23 × 3 × 337 is the 7071th composite number and is not squarefree.
8088 has 3 distinct prime factors, 16 divisors, 9 antidivisors and 2688 totatives.
8088 = 20232 - 20212 = 10132 - 10092 = 6772 - 6712 = 3432 - 3312 is the difference of 2 nonnegative squares in 4 ways.
8088 is the sum of 2 positive triangular numbers.
8088 = 42 + 262 + 862 is the sum of 3 positive squares.
80882 = 42002 + 69122 is the sum of 2 positive squares in 1 way.
80882 is the sum of 3 positive squares.
8088 is a proper divisor of 6732 - 1.
8088 is palindromic in (at least) the following bases: 23, 43, 49, -25, -47, and -55.
8088 consists of only the digits '0' and '8'.
8088 in base 23 = f6f and consists of only the digits '6' and 'f'.
8088 in base 42 = 4OO and consists of only the digits '4' and 'O'.
8088 in base 43 = 4G4 and consists of only the digits '4' and 'G'.
8088 in base 48 = 3OO and consists of only the digits '3' and 'O'.
8088 in base 49 = 3I3 and consists of only the digits '3' and 'I'.

The number 8088 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A006951: Number of conjugacy classes in GL(n,2).
A035937: Number of partitions in parts not of the form 7k, 7k+1 or 7k-1. Also number of partitions with no part of size 1 and differences between parts at distance 2 are greater than 1.
A084832: Numbers n such that 2*R_n - 1 is prime, where R_n = 11...1 is the repunit (A002275) of length n.
A101198: Number of partitions of n with rank 1 (the rank of a partition is the largest part minus the number of parts).
A114615: Starting numbers for which the RATS sequence has eventual period 14.
A204095: Numbers whose set of base 10 digits is {0,8}.
A266493: T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with no element 1 greater than its north, west, northwest or northeast neighbor modulo n and the upper left element equal to 0.
A274471: Numbers missing from A134419 despite satisfying the necessary congruence conditions (see comments).
A292738: Numbers in which 8 outnumbers all other digits together.
A293100: T(n,k)=Number of nXk 0..1 arrays with the number of 1s horizontally, diagonally or antidiagonally adjacent to some 0 two less than the number of 0s adjacent to some 1.

Saturday, April 21, 2018

Number of the day: 60509

Properties of the number 60509:

60509 is a cyclic number.
60509 is the 6101th prime.
60509 has 15 antidivisors and 60508 totatives.
60509 has an oblong digit sum 20 in base 10.
60509 = 302552 - 302542 is the difference of 2 nonnegative squares in 1 way.
60509 is the sum of 2 positive triangular numbers.
60509 is the difference of 2 positive pentagonal numbers in 1 way.
60509 = 222 + 2452 is the sum of 2 positive squares in 1 way.
60509 = 102 + 532 + 2402 is the sum of 3 positive squares.
605092 = 107802 + 595412 is the sum of 2 positive squares in 1 way.
605092 is the sum of 3 positive squares.
60509 is a proper divisor of 532161 - 1.
60509 is an emirp in (at least) the following bases: 4, 5, 6, 22, 26, 33, 38, 40, 44, 47, 58, 62, 67, 73, 77, 83, and 92.
60509 is palindromic in (at least) the following bases: 45, and 51.
60509 in base 45 = TdT and consists of only the digits 'T' and 'd'.
60509 in base 51 = NDN and consists of only the digits 'D' and 'N'.

The number 60509 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A018847: Strobogrammatic primes: the same upside down (calculator-style numerals).
A028883: Primes of form n^2 - 7.
A105378: Primes from merging of 5 successive digits in decimal expansion of Zeta(2) or (Pi^2)/6.
A201487: Primes of the form 5n^2 + 9.
A242385: Lesser of consecutive primes such that their average is of the form k*(k+2), for some integer k.
A242991: Smallest prime p such that p - floor(sqrt(p))^2 = n^2.

Friday, April 20, 2018

Number of the day: 291884

Properties of the number 291884:

291884 = 22 × 43 × 1697 is the 266506th composite number and is not squarefree.
291884 has 3 distinct prime factors, 12 divisors, 11 antidivisors and 142464 totatives.
291884 = 729722 - 729702 = 17402 - 16542 is the difference of 2 nonnegative squares in 2 ways.
291884 is the difference of 2 positive pentagonal numbers in 2 ways.
291884 = 702 + 782 + 5302 is the sum of 3 positive squares.
2918842 = 564162 + 2863802 is the sum of 2 positive squares in 1 way.
2918842 is the sum of 3 positive squares.
291884 is a proper divisor of 128312 - 1.