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%I A351982 #10 Mar 29 2022 02:28:20
%S A351982 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 A351982 10,17,14,14,12,17,17,21,18,23,20,28,27,31,27,36,32,35,37,46,41,52,45,
%U A351982 60,58,63,59,78,71,76,81,87,80,103,107,113,114,127
%N A351982 Number of integer partitions of n into prime parts with prime multiplicities.
%e A351982 The partitions for n = 4, 6, 10, 19, 20, 25:
%e A351982   (22)  (33)   (55)     (55333)     (7733)       (55555)
%e A351982         (222)  (3322)   (55522)     (77222)      (77722)
%e A351982                (22222)  (3333322)   (553322)     (5533333)
%e A351982                         (33322222)  (5522222)    (5553322)
%e A351982                                     (332222222)  (55333222)
%e A351982                                                  (55522222)
%e A351982                                                  (333333322)
%e A351982                                                  (3333322222)
%t A351982 Table[Length[Select[IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]
%Y A351982 The version for just prime parts is A000607, ranked by A076610.
%Y A351982 The version for just prime multiplicities is A055923, ranked by A056166.
%Y A351982 For odd instead of prime we have A117958, ranked by A352142.
%Y A351982 The constant case is A230595, ranked by A352519.
%Y A351982 Allowing any multiplicity > 1 gives A339218, ranked by A352492.
%Y A351982 These partitions are ranked by A346068.
%Y A351982 The non-constant case is A352493, ranked by A352518.
%Y A351982 A000040 lists the primes.
%Y A351982 A001221 counts constant partitions of prime length, ranked by A053810.
%Y A351982 A001694 lists powerful numbers, counted A007690, weak A052485.
%Y A351982 A038499 counts partitions of prime length.
%Y A351982 A101436 counts parts of prime signature that are themselves prime.
%Y A351982 A112798 lists prime indices, reverse A296150, sum A056239.
%Y A351982 A124010 gives prime signature, sorted A118914, sum A001222.
%Y A351982 A257994 counts prime indices that are prime, nonprime A330944.
%Y A351982 Cf. A000720, A000961, A001156, A011757, A052335, A164336, A320628.
%K A351982 nonn
%O A351982 0,7
%A A351982 _Gus Wiseman_, Mar 18 2022

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