A378222 Number of ordered factorizations of the odd part of n into factors > 1.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 3, 1, 1, 3, 1, 3, 2, 1, 1, 3, 1, 1, 3, 1, 1, 8, 1, 1, 1, 2, 2, 3, 1, 1, 4, 3, 1, 3, 1, 1, 3, 1, 1, 8, 1, 3, 3, 1, 1, 3, 3, 1, 2, 1, 1, 8, 1, 3, 3, 1, 1, 8, 1, 1, 3, 3, 1, 3, 1, 1, 8, 3, 1, 3, 1, 3, 1, 1, 2, 8, 2, 1, 3, 1, 1, 13

Offset: 1

Author

Antti Karttunen, Nov 24 2024

Keywords

• nonn
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000265, A002033, A074206.

Bisections: A352063, and the sequence itself.

Programs

Mathematica

f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n &]; a[n_] := f[n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, May 01 2025 *)

PARI

A000265(n) = (n>>valuation(n,2));
A074206(n) = if(n>1, sumdiv(n, i, if(iA074206(i))), n); \\ From A074206
A378222(n) = A074206(A000265(n));

Formula

a(n) = A074206(A000265(n)).

For all n >= 1, a(2n) = a(n).

For all n >= 0, a(2n+1) = A352063(n) = A002033(2*n) = A074206(2*n+1).