%I #10 Mar 29 2022 02:28:20

%S 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,

%T 10,17,14,14,12,17,17,21,18,23,20,28,27,31,27,36,32,35,37,46,41,52,45,

%U 60,58,63,59,78,71,76,81,87,80,103,107,113,114,127

%N Number of integer partitions of n into prime parts with prime multiplicities.

%e The partitions for n = 4, 6, 10, 19, 20, 25:

%e (22) (33) (55) (55333) (7733) (55555)

%e (222) (3322) (55522) (77222) (77722)

%e (22222) (3333322) (553322) (5533333)

%e (33322222) (5522222) (5553322)

%e (332222222) (55333222)

%e (55522222)

%e (333333322)

%e (3333322222)

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

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

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

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

%Y The constant case is A230595, ranked by A352519.

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

%Y These partitions are ranked by A346068.

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

%Y A000040 lists the primes.

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

%Y A001694 lists powerful numbers, counted A007690, weak A052485.

%Y A038499 counts partitions of prime length.

%Y A101436 counts parts of prime signature that are themselves prime.

%Y A112798 lists prime indices, reverse A296150, sum A056239.

%Y A124010 gives prime signature, sorted A118914, sum A001222.

%Y A257994 counts prime indices that are prime, nonprime A330944.

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

%K nonn

%O 0,7

%A _Gus Wiseman_, Mar 18 2022