fewerfactor

Fewerfactor is a concept in number theory used to refer to the number of distinct prime factors that occur with an odd exponent in the prime factorization of a positive integer. If n > 1 and n = product over i of p_i^{a_i} is the prime factorization of n, then the fewerfactor of n, denoted F(n), is defined by F(n) = the count of indices i for which a_i is odd. By convention, F(1) = 0.

Basic properties: F(n) ranges from 0 to ω(n), where ω(n) is the number of distinct prime divisors of n. F(n) = 0 precisely when n is a perfect square. For any positive integers m and n, F(mn) ≡ F(m) + F(n) (mod 2). The value therefore reflects the parity pattern of the exponents in the factorization rather than the exponents themselves.

Examples: F(72) = 1 because 72 = 2^3 · 3^2; F(24) = 2 since 24 = 2^3 · 3; and F(6) = 2 since 6 = 2 · 3. Relationship to related notions: F(n) is connected to the parity of the exponent vector and to the square-free kernel rad(n), since it counts how many primes divide n with odd exponent. As a simple parity invariant, it can aid in rapid factorization checks or in educational explanations of how exponents influence factorization structure.

History and usage: The term and definition appear in expository discussions of parity properties in factorizations and are not part of a single standardized formal object; variations may define F(n) differently in specific contexts.

See also: prime factorization, ω(n), rad(n), square-free numbers, parity function.