A324748 Number of strict integer partitions of n containing all prime indices of the parts.

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, 22, 20, 28, 31, 32, 36, 41, 43, 53, 53, 59, 70, 76, 77, 89, 99, 108, 124, 135, 139, 160, 172, 188, 209, 229, 243, 274, 298, 315, 353, 391, 417, 457, 496, 538, 588

Offset: 0

Author

Gus Wiseman, Mar 15 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.

Examples

			The first 15 terms count the following integer partitions.
   1: (1)
   3: (2,1)
   5: (4,1)
   6: (3,2,1)
   7: (4,2,1)
   9: (8,1)
   9: (6,2,1)
  10: (4,3,2,1)
  11: (8,2,1)
  11: (5,3,2,1)
  12: (9,2,1)
  12: (7,4,1)
  12: (6,3,2,1)
  13: (8,4,1)
  13: (6,4,2,1)
  14: (8,3,2,1)
  14: (7,4,2,1)
  15: (12,2,1)
  15: (9,3,2,1)
  15: (8,4,2,1)
  15: (5,4,3,2,1)
An example for n = 6 is (20,18,11,5,3,2,1), with prime indices:
  20: {1,1,3}
  18: {1,2,2}
  11: {5}
   5: {3}
   3: {2}
   2: {1}
   1: {}
All of these prime indices {1,2,3,5} belong to the partition, as required.
		

Crossrefs

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

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

Cf. A324697, A324737, A324751.

Programs

Mathematica

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