A385416 The number of unordered factorizations of n into exponentially odd numbers (A268335).

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 2, 4, 2, 5, 1, 2, 2, 5, 1, 5, 1, 2, 2, 2, 2, 5, 1, 5, 2, 2, 1, 6, 2, 2, 2

Offset: 1

Author

Amiram Eldar, Jun 28 2025

Keywords

• nonn

Comments

a(n) depends only on the prime signature of n (A118914).

The number of unordered factorizations of n into exponentially odd prime powers (A246551) is A050361(n).

Examples

			a(6) = 2 since 6 has 2 factorizations: 2 * 3 and 6.
a(24) = 4 since 24 has 4 factorizations: 2 * 2 * 2 * 3, 2 * 2 * 6, 3 * 8, and 24.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Unordered Factorization.

Crossrefs

Cf. A001055, A005117, A050361, A118914, A246551, A268335, A385417 (indices of records).

Programs

Mathematica

T[n_, k_] := T[n, k] = If[n == 1 || k == 1, {{}}, Flatten[Table[Map[Prepend[#, d] &, T[d, k/d]], {d, Select[Divisors[k], 1 < # <= n &]}], 1]]; f[n_] := T[n, n];
expOdd[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; a[n_] := Count[f[n], _?(And @@ (expOdd /@ #) &)]; Array[a, 100]

Formula

a(n) <= A001055(n), with equality if and only if n is squarefree (A005117).