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

%I A385416 #8 Jun 29 2025 10:09:44
%S A385416 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,
%T A385416 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,
%U A385416 2,5,1,5,1,2,2,2,2,5,1,5,2,2,1,6,2,2,2
%N A385416 The number of unordered factorizations of n into exponentially odd numbers (A268335).
%C A385416 a(n) depends only on the prime signature of n (A118914).
%C A385416 The number of unordered factorizations of n into exponentially odd prime powers (A246551) is A050361(n).
%H A385416 Amiram Eldar, <a href="/A385416/b385416.txt">Table of n, a(n) for n = 1..10000</a>
%H A385416 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnorderedFactorization.html">Unordered Factorization</a>.
%F A385416 a(n) <= A001055(n), with equality if and only if n is squarefree (A005117).
%e A385416 a(6) = 2 since 6 has 2 factorizations: 2 * 3 and 6.
%e A385416 a(24) = 4 since 24 has 4 factorizations: 2 * 2 * 2 * 3, 2 * 2 * 6, 3 * 8, and 24.
%t A385416 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];
%t A385416 expOdd[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; a[n_] := Count[f[n], _?(And @@ (expOdd /@ #) &)]; Array[a, 100]
%Y A385416 Cf. A001055, A005117, A050361, A118914, A246551, A268335, A385417 (indices of records).
%K A385416 nonn
%O A385416 1,6
%A A385416 _Amiram Eldar_, Jun 28 2025