A351982 Number of integer partitions of n into prime parts with prime multiplicities.

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 3, 0, 1, 1, 3, 3, 3, 0, 1, 4, 5, 5, 3, 3, 5, 8, 5, 5, 6, 8, 8, 11, 7, 8, 10, 17, 14, 14, 12, 17, 17, 21, 18, 23, 20, 28, 27, 31, 27, 36, 32, 35, 37, 46, 41, 52, 45, 60, 58, 63, 59, 78, 71, 76, 81, 87, 80, 103, 107, 113, 114, 127

Offset: 0

Author

Gus Wiseman, Mar 18 2022

Keywords

• nonn

Examples

			The partitions for n = 4, 6, 10, 19, 20, 25:
  (22)  (33)   (55)     (55333)     (7733)       (55555)
        (222)  (3322)   (55522)     (77222)      (77722)
               (22222)  (3333322)   (553322)     (5533333)
                        (33322222)  (5522222)    (5553322)
                                    (332222222)  (55333222)
                                                 (55522222)
                                                 (333333322)
                                                 (3333322222)
		

Crossrefs

The version for just prime parts is A000607, ranked by A076610.

The version for just prime multiplicities is A055923, ranked by A056166.

For odd instead of prime we have A117958, ranked by A352142.

The constant case is A230595, ranked by A352519.

Allowing any multiplicity > 1 gives A339218, ranked by A352492.

These partitions are ranked by A346068.

The non-constant case is A352493, ranked by A352518.

A000040 lists the primes.

A001221 counts constant partitions of prime length, ranked by A053810.

A001694 lists powerful numbers, counted A007690, weak A052485.

A038499 counts partitions of prime length.

A101436 counts parts of prime signature that are themselves prime.

A112798 lists prime indices, reverse A296150, sum A056239.

A124010 gives prime signature, sorted A118914, sum A001222.

A257994 counts prime indices that are prime, nonprime A330944.

Cf. A000720, A000961, A001156, A011757, A052335, A164336, A320628.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]