%I #6 May 17 2024 19:47:26
%S 0,0,2,2,1,3,3,6,3,6,9,20,13,22,22,45,47,70,75,100,107,132,157,202,
%T 229,302,396,495,536,699,820,962,1193,1507,1699,2064,2455,2945,3408,
%U 4026,4691,5749,6670,7614,9127,10930,12329,14370,16955,19961,22950,26574,30941
%N Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.
%C Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).
%e The partition (3,2,1) has rank 2^(3-1) + 2^(2-1) + 2^(1-1) = 7, which is prime, so (3,2,1) is counted under a(6).
%e The a(2) = 2 through a(10) = 9 partitions:
%e (2) (21) (31) (221) (51) (421) (431) (441) (91)
%e (11) (111) (2111) (321) (2221) (521) (3321) (631)
%e (11111) (3111) (4111) (5111) (4221) (721)
%e (22111) (33111) (3331)
%e (211111) (42111) (7111)
%e (1111111) (411111) (32221)
%e (322111)
%e (3211111)
%e (31111111)
%t Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^#]/2]&]],{n,0,30}]
%Y For all positive integers (not just prime) we get A000041.
%Y For even instead of prime we have A087787, strict A025147, odd A096765.
%Y These partitions have Heinz numbers A277319.
%Y The strict case is A372687, ranks A372851.
%Y The version counting only distinct parts is A372887, ranks A372850.
%Y A014499 lists binary indices of prime numbers.
%Y A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A048793 and A272020 (reverse) list binary indices:
%Y - length A000120
%Y - min A001511
%Y - sum A029931
%Y - max A070939
%Y A058698 counts partitions of prime numbers, strict A064688.
%Y A372885 lists primes whose binary indices sum to a prime, indices A372886.
%Y Cf. A000040, A005940, A023506, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372689.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 16 2024