|
%I #20 Jan 11 2020 20:48:59
%S 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,
%T 57,64,66,67,74,77,92,97,98,101,105,109,118,135,137,139,141,162,165,
%U 171,176,181,189,195,198,203,212,231,249,250,254,257,267,269,272,289
%N Number of factorizations of n for some n (image of A001055).
%C Also the image of A318284. - _Gus Wiseman_, Jan 11 2020
%H Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, <a href="http://arxiv.org/abs/0807.0986">On the Oppenheim's "factorisatio numerorum" function</a>, arXiv:0807.0986 [math.NT], 2008.
%F 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
%t terms = 61; m0 = 10^5; dm = 10^4;
%t f[1, _] = 1; f[n_, k_] := f[n, k] = Sum[f[n/d, d], {d, Select[Divisors[n], 1 < # <= k &]}];
%t 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];
%t seq[m] (* _Jean-François Alcover_, Oct 04 2018 *)
%Y Factorizations are A001055 with image this sequence and complement A330976.
%Y Strict factorizations are A045778 with image A045779 and complement A330975.
%Y The least number with exactly a(n) factorizations is A045783(n).
%Y The least number with exactly n factorizations is A330973(n).
%Y Cf. A002033, A007716, A033833, A318284, A325238, A330935, A330936, A330977, A330989, A330991, A330992, A330997.
%K nonn
%O 1,2
%A _David W. Wilson_
%E Name edited by _Gus Wiseman_, Jan 11 2020
|
