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Numbers that are not the number of factorizations of n into distinct factors > 1 for any n.
12

%I #11 Jan 26 2020 20:42:45

%S 11,13,20,23,24,26,28,29,30,35,36,37,39,41,45,47,48,49,50,51,53,58,60,

%T 62,63,65,66,68,69,71,72,73,75,77,78,79,81,82,84,85,86,87,90,92,94,95,

%U 96,97,98,99,101,102,103,105,106,107,108,109,113,114,115,118

%N Numbers that are not the number of factorizations of n into distinct factors > 1 for any n.

%C Warning: I have only confirmed the first three terms. The rest are derived from A045779. - _Gus Wiseman_, Jan 07 2020

%H R. E. Canfield, P. Erdős and C. Pomerance, <a href="http://math.dartmouth.edu/~carlp/PDF/paper39.pdf">On a Problem of Oppenheim concerning "Factorisatio Numerorum"</a>, J. Number Theory 17 (1983), 1-28.

%t nn=20;

%t fam[n_]:=fam[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[fam[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t nds=Length/@Array[Select[fam[#],UnsameQ@@#&]&,2^nn];

%t Complement[Range[nn],nds]

%Y Complement of A045779.

%Y The non-strict version is A330976.

%Y Factorizations are A001055, with image A045782, with complement A330976.

%Y Strict factorizations are A045778, with image A045779.

%Y The least positive integer with n strict factorizations is A330974(n).

%Y Cf. A001222, A002033, A025487, A033833, A045780, A045783, A318286, A328966, A330972, A330973, A330997.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 07 2020