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A324755
Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.
9
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
0,5
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !MemberQ[#, k_/; SubsetQ[#, PrimePi/@First/@If[k==1, {}, FactorInteger[k]]]]&]], {n, 0, 30}]
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.
Sequence in context: A269595 A055176 A118267 * A259019 A375779 A262663
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 16 2019
STATUS
approved