A371292 Numbers whose binary indices have prime indices covering an initial interval of positive integers.

0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 86, 87, 92, 93, 94, 95, 112, 113, 114, 115, 116, 117, 118, 119

Offset

0,3

Comments

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.

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.

Links

Table of n, a(n) for n=0..65.

Example

The terms together with their prime indices of binary indices begin:
   0: {}
   1: {{}}
   2: {{1}}
   3: {{},{1}}
   6: {{1},{2}}
   7: {{},{1},{2}}
   8: {{1,1}}
   9: {{},{1,1}}
  10: {{1},{1,1}}
  11: {{},{1},{1,1}}
  12: {{2},{1,1}}
  13: {{},{2},{1,1}}
  14: {{1},{2},{1,1}}
  15: {{},{1},{2},{1,1}}
  22: {{1},{2},{3}}
  23: {{},{1},{2},{3}}
  28: {{2},{1,1},{3}}
  29: {{},{2},{1,1},{3}}
  30: {{1},{2},{1,1},{3}}
  31: {{},{1},{2},{1,1},{3}}
  32: {{1,2}}

Mathematica

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], normQ[Join@@prix/@bpe[#]]&]

Crossrefs

The case with squarefree product of prime indices is A371293.

For binary indices of each prime index we have A371447, A371448.

The connected components of this multiset system are counted by A371452.

A000009 counts partitions covering initial interval, compositions A107429.

A000670 counts patterns, ranked by A333217.

A011782 counts multisets covering an initial interval.

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

A070939 gives length of binary expansion.

A131689 counts patterns by number of distinct parts.

Cf. A000040, A001222, A055887, A255906, A326782, A368109, A371291, A371294.

Sequence in context: A225755 A126388 A039247 * A039189 A039141 A008541

Adjacent sequences: A371289 A371290 A371291 * A371293 A371294 A371295

Keyword

nonn,base

Author

Gus Wiseman, Mar 27 2024

Status

approved