A372431 Positive integers k such that the prime indices of k are disjoint from the binary indices of k.

1, 2, 4, 7, 8, 9, 10, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 26, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 53, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 84, 86, 89, 92, 93, 94, 96, 97, 98, 101

Offset: 1

Author

Gus Wiseman, May 03 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.

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.

Examples

			The binary indices of 65 are {1,7}, and the prime indices are {3,6}, so 65 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     7: {4}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    13: {6}
    16: {1,1,1,1}
The terms together with their binary expansions and binary indices begin:
   1:       1 ~ {1}
   2:      10 ~ {2}
   4:     100 ~ {3}
   7:     111 ~ {1,2,3}
   8:    1000 ~ {4}
   9:    1001 ~ {1,4}
  10:    1010 ~ {2,4}
  11:    1011 ~ {1,2,4}
  12:    1100 ~ {3,4}
  13:    1101 ~ {1,3,4}
  16:   10000 ~ {5}
		

Crossrefs

For subset instead of disjoint we have A372430.

The complement is A372432.

Equal lengths: A071814, zeros of A372441.

Equal sums: A372427, zeros of A372428.

Equal maxima: A372436, zeros of A372442.

A019565 gives Heinz number of binary indices, adjoint A048675.

A029837 gives greatest binary index, least A001511.

A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

A061395 gives greatest prime index, least A055396.

A070939 gives length of binary expansion.

A112798 lists prime indices, length A001222, reverse A296150, sum A056239.

Cf. A000720, A001221, A059893, A096111, A230877, A243055, A304818, A355536, A358136, A372429.

Programs

Mathematica

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[bix[#],prix[#]]=={}&]