A370809 - OEIS

%I #9 Sep 17 2024 12:34:09

%S 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 10,9,12,10,12,12,12,12,16,13,16,16,18,16,20,18,20,20,24,20,24,24,24,

%U 26,30,26,30,30,32,32,36,32,36,36,40,38,42,40,45,44,48

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

%e 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 For the partitions of 6 we have the following choices:

%e   (6): {{2},{3}}

%e   (51): {}

%e   (42): {{2,2}}

%e   (411): {}

%e   (33): {{3,3}}

%e   (321): {}

%e   (3111): {}

%e   (222): {{2,2,2}}

%e   (2211): {}

%e   (21111): {}

%e   (111111): {}

%e So a(6) = 2.

%t Table[Max[Length[Union[Sort /@ Tuples[If[#==1,{},First/@FactorInteger[#]]& /@ #]]]&/@IntegerPartitions[n]],{n,0,30}]

%Y For just all divisors (not just prime factors) we have A370808.

%Y The version for factorizations is A370817, for all divisors A370816.

%Y A000041 counts integer partitions, strict A000009.

%Y A006530 gives greatest prime factor, least A020639.

%Y A027746 lists prime factors, A112798 indices, length A001222.

%Y A355741, A355744, A355745 choose prime factors of prime indices.

%Y A368413 counts non-choosable factorizations, complement A368414.

%Y A370320 counts non-condensed partitions, ranks A355740.

%Y A370592, A370593, A370594, `A370807 count non-choosable partitions.

%Y Cf. A000792, A048249, A063834, A239312, A319055, A339095, A355529, A355733, A367771, A368100, A370585.

%K nonn

%O 0,7

%A _Gus Wiseman_, Mar 05 2024

%E Terms a(31) onward from _Max Alekseyev_, Sep 17 2024