singleprimefactor

Singleprimefactor refers to the set of positive integers whose prime factorization involves only one distinct prime. Equivalently, these are the prime powers of the form p^k where p is a prime and k is a positive integer. This includes all primes (k = 1) and higher powers such as 4, 8, 9, 16, 25, 27, 32, and so on.

Notation and characterization: The set can be written as S = { p^k : p prime, k ≥ 1 }. An integer n belongs to S exactly when n = p^k for some prime p and integer k ≥ 1. Since there is a single prime in its factorization, there is a unique base prime associated with each element of S.

Examples: 2, 3, 5, 7, 11, 13 (primes); 4 = 2^2, 8 = 2^3, 9 = 3^2, 16 = 2^4, 27 = 3^3, 32 = 2^5, 49 = 7^2, etc.

Properties: These numbers have exactly one distinct prime in their prime factorization. They are squarefree if and only if k = 1 (i.e., the primes themselves). For n = p^k, the number of divisors is k + 1. If k ≥ 2, the number is a powerful number, since the prime appears with exponent at least 2.

Counting and distribution: Let S(x) be the count of elements of S not exceeding x. Then S(x) = sum_{p ≤ x} floor(log_p x). In particular, primes contribute π(x), and higher powers contribute additional, smaller terms; thus S(x) ~ x / log x as x grows large, with higher powers forming a lower-order term.

See also: prime powers, prime factorization, powerful numbers, divisor function.