cp's OEIS Frontend

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.

%I A372687 #9 May 17 2024 19:49:17
%S A372687 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,
%T A372687 19,46,33,38,49,65,47,67,83,92,94,113,103,130,146,127,215,224,176,234,
%U A372687 306,270,357,383,339,393,537,540,597,683,576,798,1026,830,1157
%N 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.
%C A372687 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.
%C A372687 Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).
%e A372687 The a(2) = 1 through a(17) = 8 prime numbers:
%e A372687   2  3  5  .  17  11  19  .  257  131  73  137  97  521  4099  1031
%e A372687               7       13     67   41       71       263  2053  523
%e A372687                              37   23       43       139  1033  269
%e A372687                                            29       83   193   163
%e A372687                                                     53   47    149
%e A372687                                                     31         101
%e A372687                                                                89
%e A372687                                                                79
%e A372687 The a(2) = 1 through a(11) = 3 strict partitions:
%e A372687   (2)  (2,1)  (3,1)  .  (5,1)    (4,2,1)  (4,3,1)  .  (9,1)    (6,4,1)
%e A372687                         (3,2,1)           (5,2,1)     (6,3,1)  (8,2,1)
%e A372687                                                       (7,2,1)  (5,3,2,1)
%t A372687 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&PrimeQ[Total[2^#]/2]&]],{n,0,30}]
%Y A372687 For all positive integers (not just prime) we get A000009.
%Y A372687 Number of prime numbers p with A029931(p) = n.
%Y A372687 For odd instead of prime we have A096765, even A025147, non-strict A087787
%Y A372687 Number of times n appears in A372429.
%Y A372687 Number of rows of A372471 with sum n.
%Y A372687 The non-strict version is A372688 (or A372887), ranks A277319 (or A372850).
%Y A372687 These (strict) partitions have Heinz numbers A372851.
%Y A372687 A014499 lists binary indices of prime numbers.
%Y A372687 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372687 A038499 counts partitions of prime length, strict A085756.
%Y A372687 A048793 lists binary indices:
%Y A372687 - length A000120
%Y A372687 - min A001511
%Y A372687 - sum A029931
%Y A372687 - max A070939
%Y A372687 - reverse A272020
%Y A372687 A058698 counts partitions of prime numbers, strict A064688.
%Y A372687 A096111 gives product of binary indices.
%Y A372687 A372689 lists numbers whose binary indices sum to a prime.
%Y A372687 A372885 lists primes whose binary indices sum to a prime, indices A372886.
%Y A372687 Cf. A000040, A005940, A023506, A029837, A035100, A071814, A230877, A231204, A359359, A372436, A372441.
%K A372687 nonn
%O A372687 0,7
%A A372687 _Gus Wiseman_, May 15 2024