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
Keywords
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.
Programs
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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]
Comments