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Number of strict integer partitions of n containing all prime indices of the parts.
10

%I #10 Aug 22 2019 08:23:49

%S 1,1,0,1,0,1,1,1,0,2,1,2,3,2,2,4,3,4,3,5,6,9,8,7,8,11,12,13,15,17,22,

%T 22,20,28,31,32,36,41,43,53,53,59,70,76,77,89,99,108,124,135,139,160,

%U 172,188,209,229,243,274,298,315,353,391,417,457,496,538,588

%N Number of strict integer partitions of n containing all prime indices of the parts.

%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.

%e The first 15 terms count the following integer partitions.

%e 1: (1)

%e 3: (2,1)

%e 5: (4,1)

%e 6: (3,2,1)

%e 7: (4,2,1)

%e 9: (8,1)

%e 9: (6,2,1)

%e 10: (4,3,2,1)

%e 11: (8,2,1)

%e 11: (5,3,2,1)

%e 12: (9,2,1)

%e 12: (7,4,1)

%e 12: (6,3,2,1)

%e 13: (8,4,1)

%e 13: (6,4,2,1)

%e 14: (8,3,2,1)

%e 14: (7,4,2,1)

%e 15: (12,2,1)

%e 15: (9,3,2,1)

%e 15: (8,4,2,1)

%e 15: (5,4,3,2,1)

%e An example for n = 6 is (20,18,11,5,3,2,1), with prime indices:

%e 20: {1,1,3}

%e 18: {1,2,2}

%e 11: {5}

%e 5: {3}

%e 3: {2}

%e 2: {1}

%e 1: {}

%e All of these prime indices {1,2,3,5} belong to the partition, as required.

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,30}]

%Y The subset version is A324736. The non-strict version is A324753. The Heinz number version is A290822. An infinite version is A324698.

%Y Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A279861, A290689, A290760, A305713.

%Y Cf. A324697, A324737, A324751.

%K nonn

%O 0,10

%A _Gus Wiseman_, Mar 15 2019