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Number of divisors of n whose prime indices cover an initial interval of positive integers.
31

%I #8 Aug 08 2022 16:02:47

%S 1,2,1,3,1,3,1,4,1,2,1,5,1,2,1,5,1,4,1,3,1,2,1,7,1,2,1,3,1,4,1,6,1,2,

%T 1,7,1,2,1,4,1,3,1,3,1,2,1,9,1,2,1,3,1,5,1,4,1,2,1,7,1,2,1,7,1,3,1,3,

%U 1,2,1,10,1,2,1,3,1,3,1,5,1,2,1,5,1,2,1

%N Number of divisors of n whose prime indices cover an initial interval of positive integers.

%C 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.

%e The a(n) gapless divisors of n = 1..24:

%e 1 2 1 4 1 6 1 8 1 2 1 12 1 2 1 16 1 18 1 4 1 2 1 24

%e 1 2 2 4 1 6 1 8 6 2 1 12

%e 1 1 2 4 4 2 1 8

%e 1 2 2 1 6

%e 1 1 4

%e 2

%e 1

%e For example, the divisors of 12 are {1,2,3,4,6,12}, of which {1,2,4,6,12} belong to A055932, so a(12) = 5.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t normQ[m_]:=m=={}||Union[m]==Range[Max[m]];

%t Table[Length[Select[Divisors[n],normQ[primeMS[#]]&]],{n,100}]

%Y These divisors belong to A055932, a subset of A073491 (complement A073492).

%Y The complement is A356225.

%Y A001223 lists the prime gaps.

%Y A328338 has third-largest divisor prime.

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

%Y Cf. A000005, A001222, A028334, A029709, A055874, A056239, A070824, A112798, A119313, A137921, A287170, A289509, A356223.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 04 2022