A372688 Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.

0, 0, 2, 2, 1, 3, 3, 6, 3, 6, 9, 20, 13, 22, 22, 45, 47, 70, 75, 100, 107, 132, 157, 202, 229, 302, 396, 495, 536, 699, 820, 962, 1193, 1507, 1699, 2064, 2455, 2945, 3408, 4026, 4691, 5749, 6670, 7614, 9127, 10930, 12329, 14370, 16955, 19961, 22950, 26574, 30941

Offset: 0

Author

Gus Wiseman, May 16 2024

Keywords

• nonn

Comments

Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

Examples

			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).
The a(2) = 2 through a(10) = 9 partitions:
(2)   (21)   (31)  (221)    (51)    (421)      (431)   (441)     (91)
(11)  (111)        (2111)   (321)   (2221)     (521)   (3321)    (631)
                   (11111)  (3111)  (4111)     (5111)  (4221)    (721)
                                    (22111)            (33111)   (3331)
                                    (211111)           (42111)   (7111)
                                    (1111111)          (411111)  (32221)
                                                                 (322111)
                                                                 (3211111)
                                                                 (31111111)
		

Crossrefs

For all positive integers (not just prime) we get A000041.

For even instead of prime we have A087787, strict A025147, odd A096765.

These partitions have Heinz numbers A277319.

The strict case is A372687, ranks A372851.

The version counting only distinct parts is A372887, ranks A372850.

A014499 lists binary indices of prime numbers.

A019565 gives Heinz number of binary indices, adjoint A048675.

A048793 and A272020 (reverse) list binary indices:

- length A000120

- min A001511

- sum A029931

- max A070939

A058698 counts partitions of prime numbers, strict A064688.

A372885 lists primes whose binary indices sum to a prime, indices A372886.

Cf. A000040, A005940, A023506, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372689.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^#]/2]&]],{n,0,30}]