A356845 Odd numbers with gapless prime indices.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 75, 77, 79, 81, 83, 89, 97, 101, 103, 105, 107, 109, 113, 121, 125, 127, 131, 135, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 191

Offset: 1

Author

Gus Wiseman, Sep 03 2022

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.

A sequence is gapless if it covers an interval of positive integers.

Examples

			The terms together with their prime indices begin:
    1: {}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}
		

Crossrefs

Consists of the odd terms of A073491.

These partitions are counted by A264396.

The strict case is A294674, counted by A136107.

The version for compositions is A356843, counted by A251729.

A001221 counts distinct prime factors, sum A001414.

A056239 adds up prime indices, row sums of A112798, lengths A001222.

A356069 counts gapless divisors, initial A356224 (complement A356225).

A356230 ranks gapless factorization lengths, firsts A356603.

A356233 counts factorizations into gapless numbers.

Cf. A003963, A034296, A055932, A073493, A107428, A287170, A289508, A325160, A356231, A356234, A356841.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[1,100,2],nogapQ[primeMS[#]]&]