divisorcount1

Divisorcount1 is not a universally defined mathematical term; in practice it is used in various contexts to denote a variant of the divisor-counting problem that applies a constraint to which divisors are included in the count. The most common interpretation is a modular constraint: divisorcount1(n) counts the divisors d of a positive integer n that satisfy d ≡ 1 (mod m) for a fixed modulus m. When m equals 2, this reduces to counting the odd divisors of n.

For example, if n = 18, the divisors are 1, 2, 3, 6, 9, 18. The odd divisors are 1, 3, and 9, so divisorcount1(18) with m = 2 equals 3. If the prime factorization of n is n = 2^a ∏ p_i^{e_i} with odd primes p_i, the number of odd divisors is ∏ (e_i + 1), since the factor 2^a contributes no odd divisors.

For general m, divisorcount1(n) requires counting divisors d of n that satisfy d ≡ 1 (mod m). There is no simple closed-form expression in general; computation typically involves factoring n, enumerating divisors, or applying modular constraints to prune possibilities. The behavior of divisorcount1WithM(n) is not multiplicative in general when m > 2, meaning its value cannot be determined solely from the prime-power components of n without considering the modular condition.

Relation to standard concepts: divisorcount1 is related to the divisor function tau(n), which counts all divisors of n, and to modular arithmetic. In software libraries, divisorcount1 may be implemented as a wrapper around divisor enumeration with an added congruence check. Users should consult the specific definition used in a given codebase or text, as interpretations can vary. See also tau(n), prime factorization, and modular arithmetic.