A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 18, 20, 24, 25, 26, 30, 32, 33, 34, 35, 36, 40, 42, 45, 47, 48, 50, 51, 52, 54, 55, 60, 64, 65, 66, 68, 70, 72, 75, 78, 80, 84, 85, 86, 90, 94, 96, 99, 100, 102, 104, 105, 108, 110, 119, 120, 123, 125, 126, 127, 128, 130

Offset

1,2

Comments

Also Heinz numbers of integer partitions whose parts have binary indices covering an initial interval.

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.

Links

Table of n, a(n) for n=1..63.

Example

The terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  18: {{1},{2},{2}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  25: {{1,2},{1,2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}

Mathematica

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
bpe[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[1000], normQ[Join@@bpe/@prix[#]]&]

Crossrefs

For prime indices of prime indices we have A320456.

For binary indices of binary indices we have A326754.

An opposite version is A371292, A371293.

The case with squarefree product of prime indices is A371448.

The connected components of this multiset system are counted by A371451.

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, reverse A272020, length A000120, sum A029931.

A070939 gives length of binary expansion.

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

A131689 counts patterns by number of distinct parts.

Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

Sequence in context: A077255 A262439 A331903 * A091413 A371448 A020640

Adjacent sequences: A371444 A371445 A371446 * A371448 A371449 A371450

Keyword

nonn

Author

Gus Wiseman, Mar 31 2024

Status

approved