A356225 Number of divisors of n whose prime indices do not cover an initial interval of positive integers.

0, 0, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 1, 2, 3, 0, 1, 2, 1, 3, 3, 2, 1, 1, 2, 2, 3, 3, 1, 4, 1, 0, 3, 2, 3, 2, 1, 2, 3, 4, 1, 5, 1, 3, 5, 2, 1, 1, 2, 4, 3, 3, 1, 3, 3, 4, 3, 2, 1, 5, 1, 2, 5, 0, 3, 5, 1, 3, 3, 6, 1, 2, 1, 2, 5, 3, 3, 5, 1, 5, 4, 2, 1, 7, 3, 2, 3

Offset: 1

Author

Gus Wiseman, Aug 13 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.

Examples

			The a(70) = 6 divisors: 5, 7, 10, 14, 35, 70.
		

Crossrefs

These divisors belong to the complement of A055932, a subset of A073491.

These divisors belong to A080259, a superset of A073492.

The complement is counted by A356224.

A001223 lists the prime gaps.

A328338 has third-largest divisor prime, smallest A119313.

A356226 gives the lengths of maximal gapless intervals of prime indices.

Cf. A000005, A001222, A028334, A055874, A056239, A070824, A112798, A137921, A287170, A356233-A356237.

Programs

Mathematica

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

Formula

a(n) = A000005(n) - A356224(n).