A372687 Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.

0, 0, 1, 1, 1, 0, 2, 1, 2, 0, 3, 3, 1, 4, 1, 6, 5, 8, 4, 12, 8, 12, 7, 20, 8, 16, 17, 27, 19, 38, 19, 46, 33, 38, 49, 65, 47, 67, 83, 92, 94, 113, 103, 130, 146, 127, 215, 224, 176, 234, 306, 270, 357, 383, 339, 393, 537, 540, 597, 683, 576, 798, 1026, 830, 1157

Offset: 0

Author

Gus Wiseman, May 15 2024

Keywords

• nonn

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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

Examples

			The a(2) = 1 through a(17) = 8 prime numbers:
  2  3  5  .  17  11  19  .  257  131  73  137  97  521  4099  1031
              7       13     67   41       71       263  2053  523
                             37   23       43       139  1033  269
                                           29       83   193   163
                                                    53   47    149
                                                    31         101
                                                               89
                                                               79
The a(2) = 1 through a(11) = 3 strict partitions:
  (2)  (2,1)  (3,1)  .  (5,1)    (4,2,1)  (4,3,1)  .  (9,1)    (6,4,1)
                        (3,2,1)           (5,2,1)     (6,3,1)  (8,2,1)
                                                      (7,2,1)  (5,3,2,1)
		

Crossrefs

For all positive integers (not just prime) we get A000009.

Number of prime numbers p with A029931(p) = n.

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

Number of times n appears in A372429.

Number of rows of A372471 with sum n.

The non-strict version is A372688 (or A372887), ranks A277319 (or A372850).

These (strict) partitions have Heinz numbers A372851.

A014499 lists binary indices of prime numbers.

A019565 gives Heinz number of binary indices, adjoint A048675.

A038499 counts partitions of prime length, strict A085756.

A048793 lists binary indices:

- length A000120

- min A001511

- sum A029931

- max A070939

- reverse A272020

A058698 counts partitions of prime numbers, strict A064688.

A096111 gives product of binary indices.

A372689 lists numbers whose binary indices sum to a prime.

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

Cf. A000040, A005940, A023506, A029837, A035100, A071814, A230877, A231204, A359359, A372436, A372441.

Programs

Mathematica

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