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A356225
Number of divisors of n whose prime indices do not cover an initial interval of positive integers.
23
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.
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.
Sequence in context: A351256 A143488 A201159 * A047070 A071127 A029381
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 13 2022
STATUS
approved