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Number of integer partitions of n containing no part > 1 whose prime indices all belong to the partition.
8

%I #8 Mar 16 2019 10:13:04

%S 1,1,2,2,4,3,7,8,11,12,19,19,30,34,46,50,71,76,104,119,151,171,225,

%T 247,315,360,446,504,629,703,867,986,1192,1346,1636,1837,2204,2500,

%U 2965,3348,3980,4475,5276,5963,6973,7852,9194,10335,12009,13536,15650,17589

%N Number of integer partitions of n containing no part > 1 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(1) = 1 through a(8) = 11 integer partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (111) (22) (311) (33) (43) (44)

%e (31) (11111) (42) (52) (62)

%e (1111) (51) (61) (71)

%e (222) (331) (422)

%e (3111) (511) (611)

%e (111111) (31111) (2222)

%e (1111111) (3311)

%e (5111)

%e (311111)

%e (11111111)

%t Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]

%Y The subset version is A324738, with maximal case A324744. The strict case is A324749. The Heinz number version is A324759. An infinite version is A324694.

%Y Cf. A000837, A001462, A007097, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324750, A324755, A324760.

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 16 2019