login
Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.
10

%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