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

Author

Gus Wiseman, Apr 20 2021

Keywords

• nonn

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.

Examples

			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}
		

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.

Programs

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]&]