factor2n

Factor2n refers to the set of positive divisors of the integer 2n, for a given positive integer n. In most contexts, the term denotes the factors or divisors of 2n rather than any separate function. The divisors of 2n are all positive integers d such that d divides 2n, and they come in complementary pairs d and 2n/d.

Prime factorization provides a practical way to determine the divisors of 2n. If n = 2^b · m, where m is odd and b ≥ 0, then 2n = 2^(b+1) · m. If m = ∏ p_i^(a_i) with odd primes p_i, the number of positive divisors of 2n is τ(2n) = (b+2) · ∏ (a_i+1). For odd n (b = 0), τ(2n) = 2 · ∏ (a_i+1). The actual divisor set is obtained by forming all products of powers of 2 up to exponent b+1 with all combinations of p_i^e_i, where 0 ≤ e_i ≤ a_i.

Example: Let n = 12 = 2^2 · 3. Then 2n = 24 = 2^3 · 3. The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24, totaling eight divisors. The divisor structure reflects the parity and the prime factors of n.

Notes: If negative divisors are included, the total count doubles. Factor2n is thus closely related to the divisor function and the prime factorization of integers, making it a common concept in elementary number theory and combinatorics.