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

2, 3, 11, 23, 29, 41, 43, 61, 71, 79, 89, 101, 103, 113, 131, 137, 149, 151, 163, 181, 191, 197, 211, 239, 269, 271, 281, 293, 307, 331, 349, 353, 373, 383, 401, 433, 457, 491, 503, 509, 523, 541, 547, 593, 641, 683, 701, 709, 743, 751, 761, 773, 827, 863, 887

Offset: 1

Author

Gus Wiseman, May 19 2024

Keywords

• nonn
• base

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.

The indices of these primes are A372886.

Examples

			The binary indices of 89 are {1,4,5,7}, with sum 17, which is prime, so 89 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    2:         10 ~ {2}
    3:         11 ~ {1,2}
   11:       1011 ~ {1,2,4}
   23:      10111 ~ {1,2,3,5}
   29:      11101 ~ {1,3,4,5}
   41:     101001 ~ {1,4,6}
   43:     101011 ~ {1,2,4,6}
   61:     111101 ~ {1,3,4,5,6}
   71:    1000111 ~ {1,2,3,7}
   79:    1001111 ~ {1,2,3,4,7}
   89:    1011001 ~ {1,4,5,7}
  101:    1100101 ~ {1,3,6,7}
  103:    1100111 ~ {1,2,3,6,7}
  113:    1110001 ~ {1,5,6,7}
  131:   10000011 ~ {1,2,8}
  137:   10001001 ~ {1,4,8}
  149:   10010101 ~ {1,3,5,8}
  151:   10010111 ~ {1,2,3,5,8}
  163:   10100011 ~ {1,2,6,8}
  181:   10110101 ~ {1,3,5,6,8}
  191:   10111111 ~ {1,2,3,4,5,6,8}
  197:   11000101 ~ {1,3,7,8}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

For prime instead of binary indices we have A006450, prime case of A316091.

Prime numbers p such that A029931(p) is also prime.

Prime case of A372689.

The indices of these primes are A372886.

A000040 lists the prime numbers, A014499 their binary indices.

A019565 gives Heinz number of binary indices, adjoint A048675.

A058698 counts partitions of prime numbers, strict A064688.

A372687 counts strict partitions of prime binary rank, counted by A372851.

A372688 counts partitions of prime binary rank, with Heinz numbers A277319.

Binary indices:

- listed A048793, sum A029931

- reversed A272020

- opposite A371572, sum A230877

- length A000120, complement A023416

- min A001511, opposite A000012

- max A070939, opposite A070940

- complement A368494, sum A359400

- opposite complement A371571, sum A359359

Cf. A000009, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372850, A372887.

Programs

Maple

filter:= proc(p)
  local L,i,t;
  L:= convert(p,base,2);
  isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, Jun 19 2025

Mathematica

Select[Range[100],PrimeQ[#] && PrimeQ[Total[First/@Position[Reverse[IntegerDigits[#,2]],1]]]&]