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A327513 Number of divisors of n that are 1, 2, or a nonprime number whose prime indices are pairwise coprime.

%I A327513 #10 Dec 07 2021 11:07:07
%S A327513 1,2,1,3,1,3,1,4,1,3,1,5,1,3,2,5,1,3,1,5,1,3,1,7,1,3,1,5,1,6,1,6,2,3,
%T A327513 2,5,1,3,1,7,1,4,1,5,2,3,1,9,1,3,2,5,1,3,2,7,1,3,1,10,1,3,1,7,1,6,1,5,
%U A327513 2,6,1,7,1,3,2,5,2,4,1,9,1,3,1,7,2,3,1,7,1,6,1,5,2,3,2,11,1,3,2,5,1,6,1,7,3
%N A327513 Number of divisors of n that are 1, 2, or a nonprime number whose prime indices are pairwise coprime.
%C A327513 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. Numbers that are 1, 2, or a nonprime number whose prime indices are pairwise coprime are listed in A302696.
%C A327513 Note that the maximum odd divisor of any entry must be squarefree.
%C A327513 Number of terms of A302696 that divide n. Put in other words, this sequence is the inverse Möbius transform of the characteristic function of A302696. - _Antti Karttunen_, Dec 06 2021
%H A327513 Antti Karttunen, <a href="/A327513/b327513.txt">Table of n, a(n) for n = 1..20000</a>
%H A327513 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>
%H A327513 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A327513 The divisors of 72 that are 1, 2, or nonprime numbers whose prime indices are pairwise coprime are: {1, 2, 4, 6, 8, 12, 24}, so a(72) = 7.
%t A327513 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A327513 Table[Length[Select[Divisors[n],#==1||CoprimeQ@@primeMS[#]&]],{n,100}]
%o A327513 (PARI)
%o A327513 isA302696(n) = if(isprimepower(n),!(n%2), if(!issquarefree(n>>valuation(n,2)), 0, my(pis=apply(primepi,factor(n)[,1])); (lcm(pis)==factorback(pis))));
%o A327513 A327513(n) = sumdiv(n,d,isA302696(d)); \\ _Antti Karttunen_, Dec 06 2021
%Y A327513 See link for additional cross-references.
%Y A327513 Cf. A000005, A056239, A112798, A302569, A302696, A304711.
%K A327513 nonn
%O A327513 1,2
%A A327513 _Gus Wiseman_, Sep 19 2019
%E A327513 Data section extended up to 105 terms by _Antti Karttunen_, Dec 06 2021