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

1,9

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..87.

Formula

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

Example

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

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}]

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.

Sequence in context: A351256 A143488 A201159 * A047070 A071127 A029381

Adjacent sequences: A356222 A356223 A356224 * A356226 A356227 A356228

Keyword

nonn

Author

Gus Wiseman, Aug 13 2022

Status

approved