cp's OEIS Frontend

A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.

%I A370809 #9 Sep 17 2024 12:34:09
%S A370809 1,0,1,1,1,1,2,1,2,2,2,2,3,2,3,3,4,3,4,4,4,4,6,4,6,6,6,6,8,6,8,8,9,8,
%T A370809 10,9,12,10,12,12,12,12,16,13,16,16,18,16,20,18,20,20,24,20,24,24,24,
%U A370809 26,30,26,30,30,32,32,36,32,36,36,40,38,42,40,45,44,48
%N A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.
%e A370809 For the partition (10,6,3,2) there are 4 choices: {2,2,2,3}, {2,2,3,3}, {2,2,3,5}, {2,3,3,5} so a(21) >= 4.
%e A370809 For the partitions of 6 we have the following choices:
%e A370809   (6): {{2},{3}}
%e A370809   (51): {}
%e A370809   (42): {{2,2}}
%e A370809   (411): {}
%e A370809   (33): {{3,3}}
%e A370809   (321): {}
%e A370809   (3111): {}
%e A370809   (222): {{2,2,2}}
%e A370809   (2211): {}
%e A370809   (21111): {}
%e A370809   (111111): {}
%e A370809 So a(6) = 2.
%t A370809 Table[Max[Length[Union[Sort /@ Tuples[If[#==1,{},First/@FactorInteger[#]]& /@ #]]]&/@IntegerPartitions[n]],{n,0,30}]
%Y A370809 For just all divisors (not just prime factors) we have A370808.
%Y A370809 The version for factorizations is A370817, for all divisors A370816.
%Y A370809 A000041 counts integer partitions, strict A000009.
%Y A370809 A006530 gives greatest prime factor, least A020639.
%Y A370809 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370809 A355741, A355744, A355745 choose prime factors of prime indices.
%Y A370809 A368413 counts non-choosable factorizations, complement A368414.
%Y A370809 A370320 counts non-condensed partitions, ranks A355740.
%Y A370809 A370592, A370593, A370594, `A370807 count non-choosable partitions.
%Y A370809 Cf. A000792, A048249, A063834, A239312, A319055, A339095, A355529, A355733, A367771, A368100, A370585.
%K A370809 nonn
%O A370809 0,7
%A A370809 _Gus Wiseman_, Mar 05 2024
%E A370809 Terms a(31) onward from _Max Alekseyev_, Sep 17 2024