A372687 - OEIS

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

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OFFSET

0,7

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).

LINKS

Table of n, a(n) for n=0..64.

EXAMPLE

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

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)

MATHEMATICA

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

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.

Sequence in context: A290537 A272569 A344788 * A068076 A138498 A276669

Adjacent sequences: A372684 A372685 A372686 * A372688 A372689 A372690

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 15 2024

STATUS

approved