cp's OEIS Frontend

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

%I A356225 #12 Jan 23 2024 16:18:08
%S A356225 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,
%T A356225 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,
%U A356225 3,6,1,2,1,2,5,3,3,5,1,5,4,2,1,7,3,2,3
%N A356225 Number of divisors of n whose prime indices do not cover an initial interval of positive integers.
%C A356225 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.
%F A356225 a(n) = A000005(n) - A356224(n).
%e A356225 The a(70) = 6 divisors: 5, 7, 10, 14, 35, 70.
%t A356225 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356225 normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t A356225 Table[Length[Select[Divisors[n],!normQ[primeMS[#]]&]],{n,100}]
%Y A356225 These divisors belong to the complement of A055932, a subset of A073491.
%Y A356225 These divisors belong to A080259, a superset of A073492.
%Y A356225 The complement is counted by A356224.
%Y A356225 A001223 lists the prime gaps.
%Y A356225 A328338 has third-largest divisor prime, smallest A119313.
%Y A356225 A356226 gives the lengths of maximal gapless intervals of prime indices.
%Y A356225 Cf. A000005, A001222, A028334, A055874, A056239, A070824, A112798, A137921, A287170, A356233-A356237.
%K A356225 nonn
%O A356225 1,9
%A A356225 _Gus Wiseman_, Aug 13 2022