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A370816 Greatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.

%I A370816 #8 Sep 17 2024 12:38:44
%S A370816 1,2,2,3,2,4,2,5,3,4,2,7,2,4,4,7,2,7,2,7,4,4,2,11,3,4,5,7,2,8,2,10,4,
%T A370816 4,4,12,2,4,4,11,2,8,2,7,7,4,2,17,3,7,4,7,2,11,4,11,4,4,2,15,2,4,7,14,
%U A370816 4,8,2,7,4,8,2,20,2,4,7,7,4,8,2,17,7,4,2
%N A370816 Greatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.
%H A370816 Max Alekseyev, <a href="/A370816/b370816.txt">Table of n, a(n) for n = 1..10000</a>
%e A370816 For the factorizations of 12 we have the following choices:
%e A370816   (2*2*3): {{1,1,1},{1,1,2},{1,1,3},{1,2,2},{1,2,3},{2,2,3}}
%e A370816     (2*6): {{1,1},{1,2},{1,3},{1,6},{2,2},{2,3},{2,6}}
%e A370816     (3*4): {{1,1},{1,2},{1,3},{1,4},{2,3},{3,4}}
%e A370816      (12): {{1},{2},{3},{4},{6},{12}}
%e A370816 So a(12) = 7.
%t A370816 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A370816 Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@facs[n]],{n,100}]
%Y A370816 The version for partitions is A370808, for just prime factors A370809.
%Y A370816 For just prime factors we have A370817.
%Y A370816 A000005 counts divisors.
%Y A370816 A001055 counts factorizations, strict A045778.
%Y A370816 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370816 A368413 counts non-choosable factorizations, complement A368414.
%Y A370816 A370813 counts non-divisor-choosable factorizations, complement A370814.
%Y A370816 Cf. A000792, A048249, A066739, A319055, A319057, A355737, A355739, A355740, A355741, A368110, A370803.
%K A370816 nonn
%O A370816 1,2
%A A370816 _Gus Wiseman_, Mar 06 2024