cp's OEIS Frontend

A372885 Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

%I A372885 #13 Jun 20 2025 10:47:03
%S A372885 2,3,11,23,29,41,43,61,71,79,89,101,103,113,131,137,149,151,163,181,
%T A372885 191,197,211,239,269,271,281,293,307,331,349,353,373,383,401,433,457,
%U A372885 491,503,509,523,541,547,593,641,683,701,709,743,751,761,773,827,863,887
%N A372885 Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.
%C A372885 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 A372885 The indices of these primes are A372886.
%H A372885 Robert Israel, <a href="/A372885/b372885.txt">Table of n, a(n) for n = 1..10000</a>
%e A372885 The binary indices of 89 are {1,4,5,7}, with sum 17, which is prime, so 89 is in the sequence.
%e A372885 The terms together with their binary expansions and binary indices begin:
%e A372885     2:         10 ~ {2}
%e A372885     3:         11 ~ {1,2}
%e A372885    11:       1011 ~ {1,2,4}
%e A372885    23:      10111 ~ {1,2,3,5}
%e A372885    29:      11101 ~ {1,3,4,5}
%e A372885    41:     101001 ~ {1,4,6}
%e A372885    43:     101011 ~ {1,2,4,6}
%e A372885    61:     111101 ~ {1,3,4,5,6}
%e A372885    71:    1000111 ~ {1,2,3,7}
%e A372885    79:    1001111 ~ {1,2,3,4,7}
%e A372885    89:    1011001 ~ {1,4,5,7}
%e A372885   101:    1100101 ~ {1,3,6,7}
%e A372885   103:    1100111 ~ {1,2,3,6,7}
%e A372885   113:    1110001 ~ {1,5,6,7}
%e A372885   131:   10000011 ~ {1,2,8}
%e A372885   137:   10001001 ~ {1,4,8}
%e A372885   149:   10010101 ~ {1,3,5,8}
%e A372885   151:   10010111 ~ {1,2,3,5,8}
%e A372885   163:   10100011 ~ {1,2,6,8}
%e A372885   181:   10110101 ~ {1,3,5,6,8}
%e A372885   191:   10111111 ~ {1,2,3,4,5,6,8}
%e A372885   197:   11000101 ~ {1,3,7,8}
%p A372885 filter:= proc(p)
%p A372885   local L,i,t;
%p A372885   L:= convert(p,base,2);
%p A372885   isprime(add(i*L[i],i=1..nops(L)))
%p A372885 end proc:
%p A372885 select(filter, [seq(ithprime(i),i=1..200)]); # _Robert Israel_, Jun 19 2025
%t A372885 Select[Range[100],PrimeQ[#] && PrimeQ[Total[First/@Position[Reverse[IntegerDigits[#,2]],1]]]&]
%Y A372885 For prime instead of binary indices we have A006450, prime case of A316091.
%Y A372885 Prime numbers p such that A029931(p) is also prime.
%Y A372885 Prime case of A372689.
%Y A372885 The indices of these primes are A372886.
%Y A372885 A000040 lists the prime numbers, A014499 their binary indices.
%Y A372885 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372885 A058698 counts partitions of prime numbers, strict A064688.
%Y A372885 A372687 counts strict partitions of prime binary rank, counted by A372851.
%Y A372885 A372688 counts partitions of prime binary rank, with Heinz numbers A277319.
%Y A372885 Binary indices:
%Y A372885 - listed A048793, sum A029931
%Y A372885 - reversed A272020
%Y A372885 - opposite A371572, sum A230877
%Y A372885 - length A000120, complement A023416
%Y A372885 - min A001511, opposite A000012
%Y A372885 - max A070939, opposite A070940
%Y A372885 - complement A368494, sum A359400
%Y A372885 - opposite complement A371571, sum A359359
%Y A372885 Cf. A000009, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372850, A372887.
%K A372885 nonn,base
%O A372885 1,1
%A A372885 _Gus Wiseman_, May 19 2024