A045782 Number of factorizations of n for some n (image of A001055).

1, 2, 3, 4, 5, 7, 9, 11, 12, 15, 16, 19, 21, 22, 26, 29, 30, 31, 36, 38, 42, 45, 47, 52, 56, 57, 64, 66, 67, 74, 77, 92, 97, 98, 101, 105, 109, 118, 135, 137, 139, 141, 162, 165, 171, 176, 181, 189, 195, 198, 203, 212, 231, 249, 250, 254, 257, 267, 269, 272, 289

Offset

1,2

Comments

Also the image of A318284. - Gus Wiseman, Jan 11 2020

Links

Table of n, a(n) for n=1..61.

Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function, arXiv:0807.0986 [math.NT], 2008.

Formula

The Luca et al. paper shows that the number of terms with a(n) <= x is x^{ O( log log log x / log log x )}. - N. J. A. Sloane, Jun 12 2009

Mathematica

terms = 61; m0 = 10^5; dm = 10^4;
f[1, _] = 1; f[n_, k_] := f[n, k] = Sum[f[n/d, d], {d, Select[Divisors[n], 1 < # <= k &]}];
Clear[seq]; seq[m_] := seq[m] = Sort[Tally[Table[f[n, n], {n, 1, m}]][[All, 1]]][[1 ;; terms]]; seq[m = m0]; seq[m += dm]; While[Print[m]; seq[m] != seq[m - dm], m += dm];
seq[m] (* Jean-François Alcover, Oct 04 2018 *)

Crossrefs

Factorizations are A001055 with image this sequence and complement A330976.

Strict factorizations are A045778 with image A045779 and complement A330975.

The least number with exactly a(n) factorizations is A045783(n).

The least number with exactly n factorizations is A330973(n).

Cf. A002033, A007716, A033833, A318284, A325238, A330935, A330936, A330977, A330989, A330991, A330992, A330997.

Sequence in context: A258456 A230843 A188551 * A184112 A064005 A306587

Adjacent sequences: A045779 A045780 A045781 * A045783 A045784 A045785

Keyword

nonn

Author

David W. Wilson

Extensions

Name edited by Gus Wiseman, Jan 11 2020

Status

approved