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Let q be a positive integer, then Gamma_0(q) is defined as the set of all matrices [a b; c d] in the modular group Gamma Gamma with c=0 (mod q). Gamma_0(q) is a subgroup of ...

Given a complete graph K_n which is two-colored, the number of forced monochromatic triangles is at least {1/3u(u-1)(u-2) for n=2u; 2/3u(u-1)(4u+1) for n=4u+1; ...

A multiple of a number x is any quantity y=nx with n an integer. If x and y are integers, then x is called a factor of y. The smallest positive number m for which there exist ...

A number n for which the product of divisors is equal to n^2. The first few are 1, 6, 8, 10, 14, 15, 21, 22, ... (OEIS A007422).

The Narayana triangle is the number triangle obtained from the Narayana numbers N(n,k), namely 1 ; 1 1 ; 1 3 1 ; 1 6 6 1 ; 1 10 20 10 1 ; 1 15 50 50 15 1 (OEIS A001263).

The series which arises in the binomial theorem for negative integer -n, (x+a)^(-n) = sum_(k=0)^(infty)(-n; k)x^ka^(-n-k) (1) = sum_(k=0)^(infty)(-1)^k(n+k-1; k)x^ka^(-n-k) ...

If m is an integer, then for every residue class r (mod m), there are infinitely many nonnegative integers n for which P(n)=r (mod m), where P(n) is the partition function P.

There are two equivalent definitions for a nilpotent matrix. 1. A square matrix whose eigenvalues are all 0. 2. A square matrix A such that A^n is the zero matrix 0 for some ...

The number of nonassociative n-products with k elements preceding the rightmost left parameter is F(n,k) = F(n-1,k)+F(n-1,k-1) (1) = (n+k-2; k)-(n+k-1; k-1), (2) where (n; k) ...

A positive value of n for which x-phi(x)=n has no solution, where phi(x) is the totient function. The first few are 10, 26, 34, 50, 52, ... (OEIS A005278).