A372689 Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.

2, 3, 4, 6, 9, 11, 12, 16, 18, 23, 26, 29, 33, 38, 41, 43, 44, 48, 50, 55, 58, 61, 64, 69, 71, 72, 74, 79, 81, 86, 89, 91, 92, 96, 101, 103, 104, 106, 111, 113, 118, 121, 131, 132, 134, 137, 142, 144, 149, 151, 152, 154, 159, 163, 164, 166, 169, 174, 176, 181

Offset

1,1

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 function taking a set s to its binary rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

Links

Table of n, a(n) for n=1..60.

Example

The terms together with their binary expansions and binary indices begin:
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   4:     100 ~ {3}
   6:     110 ~ {2,3}
   9:    1001 ~ {1,4}
  11:    1011 ~ {1,2,4}
  12:    1100 ~ {3,4}
  16:   10000 ~ {5}
  18:   10010 ~ {2,5}
  23:   10111 ~ {1,2,3,5}
  26:   11010 ~ {2,4,5}
  29:   11101 ~ {1,3,4,5}
  33:  100001 ~ {1,6}
  38:  100110 ~ {2,3,6}
  41:  101001 ~ {1,4,6}
  43:  101011 ~ {1,2,4,6}
  44:  101100 ~ {3,4,6}
  48:  110000 ~ {5,6}
  50:  110010 ~ {2,5,6}
  55:  110111 ~ {1,2,3,5,6}
  58:  111010 ~ {2,4,5,6}
  61:  111101 ~ {1,3,4,5,6}

Mathematica

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

Crossrefs

Numbers k such that A029931(k) is prime.

Union of prime-indexed rows of A118462.

For even instead of prime we have A158704, odd A158705.

For prime indices instead of binary indices we have A316091.

The prime case is A372885, indices 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.

A372471 lists binary indices of primes, row-sums A372429.

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

A372689 lists numbers whose binary indices sum to a prime.

A372885 lists primes whose binary indices sum to a prime, indices A372886.

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. A035100, A071814, A096111, A023506, A277319, A372688, A372850, A372887.

Sequence in context: A163627 A189887 A179970 * A018431 A184350 A347657

Adjacent sequences: A372686 A372687 A372688 * A372690 A372691 A372692

Keyword

nonn,base

Author

Gus Wiseman, May 18 2024

Status

approved