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An NSW number (named after Newman, Shanks, and Williams) is an integer m that solves the Diophantine equation 2n^2=m^2+1. (1) In other words, the NSW numbers m index the ...

Stirling's approximation gives an approximate value for the factorial function n! or the gamma function Gamma(n) for n>>1. The approximation can most simply be derived for n ...

Let sopfr(n) be the sum of prime factors (with repetition) of a number n. For example, 20=2^2·5, so sopfr(20)=2+2+5=9. Then sopfr(n) for n=1, 2, ... is given by 0, 2, 3, 4, ...

A "weird number" is a number that is abundant (i.e., the sum of proper divisors is greater than the number) without being pseudoperfect (i.e., no subset of the proper ...

A run is a sequence of more than one consecutive identical outcomes, also known as a clump. Let R_p(r,n) be the probability that a run of r or more consecutive heads appears ...

The number 10 (ten) is the basis for the decimal system of notation. In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied ...

An acyclic digraph is a directed graph containing no directed cycles, also known as a directed acyclic graph or a "DAG." Every finite acyclic digraph has at least one node of ...

Let B_k be the kth Bernoulli number and consider nB_(n-1)=-1 (mod n), where the residues of fractions are taken in the usual way so as to yield integers, for which the ...

Define a Bouniakowsky polynomial as an irreducible polynomial f(x) with integer coefficients, degree >1, and GCD(f(1),f(2),...)=1. The Bouniakowsky conjecture states that ...

Determining the maximum number of pieces in which it is possible to divide a circle for a given number of cuts is called the circle cutting or pancake cutting problem. The ...