A372850 Numbers whose distinct prime indices are the binary indices of some prime number.

3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 30, 36, 40, 42, 44, 46, 48, 50, 54, 60, 66, 70, 72, 80, 81, 84, 88, 90, 92, 96, 100, 102, 108, 114, 118, 120, 126, 130, 132, 140, 144, 150, 160, 162, 168, 176, 180, 182, 184, 192, 198, 200, 204, 216, 228, 236, 238, 240, 242

Offset: 1

Author

Gus Wiseman, May 16 2024

Keywords

• nonn

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.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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

Examples

			The distinct prime indices of 45 are {2,3}, which are the binary indices of 6, which is not prime, so 45 is not in the sequence.
The distinct prime indices of 60 are {1,2,3}, which are the binary indices of 7, which is prime, so 60 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
		

Crossrefs

For even instead of prime we have A005408, with multiplicity A003159.

For odd instead of prime we have A005843, with multiplicity A036554.

For prime indices with multiplicity we have A277319, counted by A372688.

The squarefree case is A372851, counted by A372687.

Partitions of this type are counted by A372887.

A014499 lists binary indices of prime numbers.

A019565 gives Heinz number of binary indices, adjoint A048675.

A038499 counts partitions of prime length, strict A085756.

A048793 and A272020 (reverse) list binary indices:

- length A000120

- min A001511

- sum A029931

- max A070939

A058698 counts partitions of prime numbers, strict A064688.

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

Cf. A000040, A005940, A023506, A029837, A035100, A071814, A096111, A372429, A372436, A372441, A372471, A372689.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeQ[Total[2^(Union[prix[#]]-1)]]&]

Formula

Numbers k such that Sum_{i:prime(i)|k} 2^(i-1) is prime, where the sum is over the distinct prime indices of k.