A339886 Numbers whose prime indices cover an interval of positive integers starting with 2.

1, 3, 9, 15, 27, 45, 75, 81, 105, 135, 225, 243, 315, 375, 405, 525, 675, 729, 735, 945, 1125, 1155, 1215, 1575, 1875, 2025, 2187, 2205, 2625, 2835, 3375, 3465, 3645, 3675, 4725, 5145, 5625, 5775, 6075, 6561, 6615, 7875, 8085, 8505, 9375, 10125, 10395, 10935

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.

Links

Table of n, a(n) for n=1..48.

Example

The sequence of terms together with their prime indices begins:
    3: {2}
    9: {2,2}
   15: {2,3}
   27: {2,2,2}
   45: {2,2,3}
   75: {2,3,3}
   81: {2,2,2,2}
  105: {2,3,4}
  135: {2,2,2,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  315: {2,2,3,4}
  375: {2,3,3,3}
  405: {2,2,2,2,3}
  525: {2,3,3,4}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  735: {2,3,4,4}
  945: {2,2,2,3,4}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Select[Range[100], normQ[primeMS[#]-1]&]

Crossrefs

The version starting at 1 is A055932.

The partitions with these Heinz numbers are counted by A264396.

Positions of 1's in A339662.

A000009 counts partitions covering an initial interval.

A000070 counts partitions with a selected part.

A016945 lists numbers with smallest prime index 2.

A034296 counts gap-free (or flat) partitions.

A056239 adds up prime indices, row sums of A112798.

A073491 lists numbers with gap-free prime indices.

A107428 counts gap-free compositions (initial: A107429).

A286469 and A286470 give greatest difference for Heinz numbers.

A325240 lists numbers with smallest prime multiplicity 2.

A342050/A342051 have prime indices with odd/even least gap.

Cf. A001223, A001522, A006128, A007052, A124010, A257989, A257993, A264401, A317090, A317589, A339737.

Sequence in context: A256388 A082897 A363523 * A147516 A233819 A131822

Adjacent sequences: A339883 A339884 A339885 * A339887 A339888 A339889

Keyword

nonn

Author

Gus Wiseman, Apr 20 2021

Status

approved