This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372887 #5 May 19 2024 19:42:46
%S A372887 0,0,1,1,3,3,6,8,12,14,21,29,36,48,56,74,94,123,144,195,235,301,356,
%T A372887 456,538,679,803,997,1189,1467,1716,2103,2488,2968,3517,4185,4907,
%U A372887 5834,6850,8032,9459,11073,12933,15130,17652,20480,24011,27851,32344,37520
%N A372887 Number of integer partitions of n whose distinct parts are the binary indices of some prime number.
%C A372887 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 A372887 Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).
%e A372887 The partition y = (4,3,1,1) has distinct parts {1,3,4}, which are the binary indices of 13, which is prime, so y is counted under a(9).
%e A372887 The a(2) = 1 through a(9) = 14 partitions:
%e A372887 (2) (21) (22) (221) (51) (331) (431) (3321)
%e A372887 (31) (311) (222) (421) (521) (4221)
%e A372887 (211) (2111) (321) (511) (2222) (4311)
%e A372887 (2211) (2221) (3221) (5211)
%e A372887 (3111) (3211) (3311) (22221)
%e A372887 (21111) (22111) (4211) (32211)
%e A372887 (31111) (5111) (33111)
%e A372887 (211111) (22211) (42111)
%e A372887 (32111) (51111)
%e A372887 (221111) (222111)
%e A372887 (311111) (321111)
%e A372887 (2111111) (2211111)
%e A372887 (3111111)
%e A372887 (21111111)
%t A372887 Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^(Union[#]-1)]]&]],{n,0,30}]
%Y A372887 For odd instead of prime we have A000041, even A002865.
%Y A372887 The strict case is A372687, ranks A372851.
%Y A372887 Counting not just distinct parts gives A372688, ranks A277319.
%Y A372887 These partitions have Heinz numbers A372850.
%Y A372887 A014499 lists binary indices of prime numbers.
%Y A372887 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372887 A058698 counts partitions of prime numbers, strict A064688.
%Y A372887 A372689 lists numbers whose binary indices sum to a prime.
%Y A372887 A372885 lists primes whose binary indices sum to a prime, indices A372886.
%Y A372887 Binary indices:
%Y A372887 - listed A048793, sum A029931
%Y A372887 - reversed A272020
%Y A372887 - opposite A371572, sum A230877
%Y A372887 - length A000120, complement A023416
%Y A372887 - min A001511, opposite A000012
%Y A372887 - max A070939, opposite A070940
%Y A372887 - complement A368494, sum A359400
%Y A372887 - opposite complement A371571, sum A359359
%Y A372887 Cf. A000040, A000041, A029837, A035100, A038499, A372429, A372471.
%K A372887 nonn
%O A372887 0,5
%A A372887 _Gus Wiseman_, May 19 2024