|
%I #6 Mar 16 2019 10:13:12
%S 1,0,1,1,2,1,4,3,5,6,10,7,16,14,23,23,35,34,53,54,75,80,112,115,160,
%T 169,223,244,315,339,442,478,604,664,832,910,1131,1245,1524,1689,2054,
%U 2263,2743,3039,3634,4042,4809,5343,6326,7035,8276,9217,10795,12011
%N Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C For example, (6,2) is such a partition because the prime indices of 6 are {1,2}, which do not all belong to the partition. On the other hand, (5,3) is not such a partition because the prime indices of 5 are {3}, and 3 belongs to the partition.
%e The a(2) = 1 through a(10) = 10 integer partitions (A = 10):
%e (2) (3) (4) (5) (6) (7) (8) (9) (A)
%e (22) (33) (43) (44) (54) (55)
%e (42) (52) (62) (63) (64)
%e (222) (422) (72) (73)
%e (2222) (333) (82)
%e (522) (433)
%e (442)
%e (622)
%e (4222)
%e (22222)
%t Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@If[k==1,{},FactorInteger[k]]]]&]],{n,0,30}]
%Y The subset version is A324739, with maximal case A324762. The strict case is A324750. The Heinz number version is A324760. An infinite version is A324694.
%Y Cf. A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324696, A324744.
%K nonn
%O 0,5
%A _Gus Wiseman_, Mar 16 2019
|
