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

Author

David W. Wilson

Keywords

• nonn

Comments

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

Links

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

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.

Programs

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 *)

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

Extensions

Name edited by Gus Wiseman, Jan 11 2020