A331024 - OEIS

A331024

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

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 7, 2, 1, 1, 9, 1, 1, 1, 5, 1, 9, 1, 3, 1, 1, 1, 10, 1, 3, 3, 5, 1, 1, 1, 5, 1

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OFFSET

1,8

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) = A330995(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}]//Denominator

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))));

A331024(n) = denominator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021

CROSSREFS

Positions of 1's include all elements of A001248 as well as A005117. The first position of a 1 that is not in A167207 is 128.

The numerators are A331023.

The rounded quotients are A331048.

The same for integer partitions is A330995.

Cf. A001055, A005117, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

Sequence in context: A326514 A077565 A316979 * A115561 A115622 A294892

Adjacent sequences: A331021 A331022 A331023 * A331025 A331026 A331027

KEYWORD

nonn,frac

AUTHOR

Gus Wiseman, Jan 08 2020

EXTENSIONS

More terms from Antti Karttunen, May 27 2021

STATUS

approved