A331023 Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 5, 1, 4, 1, 4, 1, 1, 1, 7, 2, 1, 3, 4, 1, 1, 1, 7, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 4, 4, 1, 1, 12, 2, 4, 1, 4, 1, 7, 1, 7, 1, 1, 1, 11, 1, 1, 4, 11, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 4, 4, 1, 1, 1, 12, 5, 1, 1, 11, 1, 1, 1, 7, 1, 11, 1, 4, 1, 1, 1, 19, 1, 4, 4, 9, 1, 1, 1, 7, 1

Offset

1,4

Comments

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different. Factorizations and strict factorizations are counted by A001055 and A045778 respectively.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

Formula

a(2^n) = A330994(n).

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[facs[n]]/Length[Select[facs[n], UnsameQ@@#&]], {n, 100}]//Numerator

Prog

(PARI)
A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));
A331023(n) = numerator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021

Crossrefs

Positions of 1's are A005117.

Positions of 2's appear to be A001248.

The denominators are A331024.

The rounded quotients are A331048.

The same for integer partitions is A330994.

Cf. A001055, A001222, A002033, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

Sequence in context: A222580 A376140 A316978 * A284345 A347463 A373378

Adjacent sequences: A331020 A331021 A331022 * A331024 A331025 A331026

Keyword

nonn,frac

Author

Gus Wiseman, Jan 08 2020

Extensions

More terms from Antti Karttunen, May 27 2021

Status

approved