A324755 Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

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, 169, 223, 244, 315, 339, 442, 478, 604, 664, 832, 910, 1131, 1245, 1524, 1689, 2054, 2263, 2743, 3039, 3634, 4042, 4809, 5343, 6326, 7035, 8276, 9217, 10795, 12011

Offset: 0

Author

Gus Wiseman, Mar 16 2019

Keywords

• nonn

Comments

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.

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.

Examples

			The a(2) = 1 through a(10) = 10 integer partitions (A = 10):
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)
            (22)       (33)   (43)  (44)    (54)   (55)
                       (42)   (52)  (62)    (63)   (64)
                       (222)        (422)   (72)   (73)
                                    (2222)  (333)  (82)
                                            (522)  (433)
                                                   (442)
                                                   (622)
                                                   (4222)
                                                   (22222)
		

Crossrefs

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.

Cf. A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324696, A324744.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@If[k==1,{},FactorInteger[k]]]]&]],{n,0,30}]