A371448 Numbers such that (1) the product of prime indices is squarefree, and (2) the binary indices of prime indices cover an initial interval of positive integers.

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 26, 30, 32, 33, 34, 40, 47, 48, 51, 52, 55, 60, 64, 66, 68, 80, 85, 86, 94, 96, 102, 104, 110, 120, 123, 127, 128, 132, 136, 141, 143, 160, 165, 170, 172, 187, 188, 192, 204, 205, 208, 215, 220, 221, 226, 240, 246

Offset

1,2

Comments

Also Heinz numbers of integer partitions whose parts have (1) squarefree product and (2) 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..60.

Formula

Intersection of A302505 and A371447.

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}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}
  33: {{2},{1,3}}
  34: {{1},{1,2,3}}
  40: {{1},{1},{1},{1,2}}
  47: {{1,2,3,4}}
  48: {{1},{1},{1},{1},{2}}
  51: {{2},{1,2,3}}

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

Crossrefs

An opposite version is A371293, A371292.

Without the squarefree condition we have A371447, see also A320456, A326754.

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: A331903 A371447 A091413 * A020640 A068383 A276603

Adjacent sequences: A371445 A371446 A371447 * A371449 A371450 A371451

Keyword

nonn

Author

Gus Wiseman, Mar 31 2024

Status

approved