A356067 Number of integer partitions of n into relatively prime prime-powers.

0, 0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 11, 7, 18, 16, 26, 27, 43, 41, 65, 65, 92, 100, 137, 142, 194, 210, 270, 295, 379, 410, 519, 571, 699, 782, 947, 1046, 1267, 1414, 1673, 1870, 2213, 2465, 2897, 3230, 3757, 4210, 4871, 5427, 6265, 6997

Offset: 0

Author

Gus Wiseman, Jul 28 2022

Keywords

• nonn

Examples

			The a(5) = 1 through a(12) = 7 partitions:
  (32)  .  (43)   (53)   (54)    (73)    (74)     (75)
           (52)   (332)  (72)    (433)   (83)     (543)
           (322)         (432)   (532)   (92)     (552)
                         (522)   (3322)  (443)    (732)
                         (3222)          (533)    (4332)
                                         (542)    (5322)
                                         (722)    (33222)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)
		

Crossrefs

This is the relatively prime case of A023894, facs A000688, w/ 1's A023893.

For strict instead of coprime: A054685, facs A050361, with 1's A106244.

The version for factorizations instead of partitions is A354911.

A000041 counts partitions, strict A000009.

A072233 counts partitions by sum and length.

A246655 lists the prime-powers (A000961 includes 1), towers A164336.

A279784 counts twice-partitions where the latter partitions are constant.

A289509 lists numbers whose prime indices are relatively prime.

A355743 lists numbers with prime-power prime indices, squarefree A356065.

Cf. A001970, A055887, A063834, A076610, A085970, A355737, A355742.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And@@PrimePowerQ/@#&&GCD@@#==1&]],{n,0,30}]