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Any nonzero rational number x can be represented by x=(p^ar)/s, (1) where p is a prime number, r and s are integers not divisible by p, and a is a unique integer. The p-adic ...

The associated Stirling numbers of the first kind d_2(n,k)=d(n,k) are defined as the number of permutations of a given number n having exactly k permutation cycles, all of ...

A composite number n is a positive integer n>1 which is not prime (i.e., which has factors other than 1 and itself). The first few composite numbers (sometimes called ...

Many algorithms have been devised for determining the prime factors of a given number (a process called prime factorization). They vary quite a bit in sophistication and ...

Multiply all the digits of a number n by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative ...

A number n is called a k e-perfect number if sigma_e(n)=kn, where sigma_e(n) is the sum of the e-divisors of n.

An amicable pair (m,n) consists of two integers m,n for which the sum of proper divisors (the divisors excluding the number itself) of one number equals the other. Amicable ...

Ramsey's theorem is a generalization of Dilworth's lemma which states for each pair of positive integers k and l there exists an integer R(k,l) (known as the Ramsey number) ...

A dozen dozen, or the square number 144.

A phi-prime is a prime number appearing in the decimal expansion of the golden ratio phi. The first few are 1618033, 1618033988749, ... (OEIS A064117). The numbers of decimal ...