Monday, February 13, 2017

Number of the day: 8512

Peter Gustav Lejeune Dirichlet was born on this day 212 years ago.

Properties of the number 8512:

8512 = 26 × 7 × 19 is the 7451th composite number and is not squarefree.
8512 has 3 distinct prime factors, 28 divisors, 15 antidivisors and 3456 totatives.
Reversing the decimal digits of 8512 results in a sphenic number.
8512 = 203 + 83 is the sum of 2 positive cubes in 1 way.
8512 is the difference of 2 nonnegative squares in 10 ways.
8512 is the difference of 2 positive pentagonal numbers in 3 ways.
8512 = 322 + 482 + 722 is the sum of 3 positive squares.
85122 is the sum of 3 positive squares.
8512 is a divisor of 12172 - 1.
8512 is palindromic in (at least) the following bases: -35, and -74.
8512 in base 23 = g22 and consists of only the digits '2' and 'g'.
8512 in base 34 = 7cc and consists of only the digits '7' and 'c'.

The number 8512 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A007764: Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid.
A037032: Total number of prime parts in all partitions of n.
A064298: Square array read by antidiagonals of self-avoiding rook paths joining opposite corners of n X k board.
A115176: Positive numbers that are not the sum of two squares and a positive Fibonacci number.
A139267: Twice octagonal numbers: 2*n*(3*n-2).
A179672: Products of the 6st power of a prime and 2 distinct primes (p^6*q*r).
A188211: T(n,k)=Number of nondecreasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero
A243953: E.g.f.: exp( Sum_{n>=1} A000108(n-1)*x^n/n ), where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.
A251088: T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock having the sum of its diagonal elements greater than the absolute difference of its antidiagonal elements
A257417: Values of n such that there are exactly 10 solutions to x^2 - y^2 = n with x > y >= 0.

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