A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.

1, 2, 3, 6, 7, 22, 23, 32, 33, 48, 49, 86, 87, 112, 113, 516, 517, 580, 581, 1110, 1111, 1136, 1137, 1604, 1605, 5206, 5207, 5232, 5233, 5700, 5701, 8212, 8213, 9236, 9237, 13332, 13333, 16386, 16387, 16450, 16451, 17474, 17475, 21570, 21571, 24576, 24577

Offset

1,2

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=1..47.

Formula

Intersection of A371292 and A371289.

Example

The terms together with their prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    3: {{},{1}}
    6: {{1},{2}}
    7: {{},{1},{2}}
   22: {{1},{2},{3}}
   23: {{},{1},{2},{3}}
   32: {{1,2}}
   33: {{},{1,2}}
   48: {{3},{1,2}}
   49: {{},{3},{1,2}}
   86: {{1},{2},{3},{4}}
   87: {{},{1},{2},{3},{4}}
  112: {{3},{1,2},{4}}
  113: {{},{3},{1,2},{4}}
  516: {{2},{1,3}}
  517: {{},{2},{1,3}}
  580: {{2},{4},{1,3}}
  581: {{},{2},{4},{1,3}}

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[1000], SquareFreeQ[Times @@ bpe[#]]&&normQ[Join@@prix/@bpe[#]]&]

Crossrefs

Without the covering condition we have A371289.

Without squarefree product we have A371292.

Interchanging binary and prime indices gives A371448.

A000009 counts partitions covering initial interval, compositions A107429.

A000670 counts ordered set partitions, allowing empty sets A000629.

A005117 lists squarefree numbers.

A011782 counts multisets covering an initial interval.

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

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

A131689 counts patterns by number of distinct parts.

A302521 lists MM-numbers of set partitions, with empties A302505.

A326701 lists BII-numbers of set partitions.

A368533 lists numbers with squarefree binary indices, prime indices A302478.

Cf. A000040, A001222, A255906, A326782, A371291, A371294, A371447, A371452.

Sequence in context: A050581 A352934 A073317 * A064731 A159069 A162681

Adjacent sequences: A371290 A371291 A371292 * A371294 A371295 A371296

Keyword

nonn

Author

Gus Wiseman, Mar 28 2024

Status

approved