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A327394 Number of stable divisors of n.

%I A327394 #13 Nov 27 2024 16:00:48
%S A327394 1,2,2,3,2,3,2,4,3,3,2,4,2,3,4,5,2,4,2,4,3,3,2,5,3,3,4,4,2,5,2,6,4,3,
%T A327394 4,5,2,3,3,5,2,4,2,4,6,3,2,6,3,4,4,4,2,5,4,5,3,3,2,6,2,3,4,7,3,5,2,4,
%U A327394 4,5,2,6,2,3,6,4,4,4,2,6,5,3,2,5,4,3,3,5,2,7,4,4,4,3,4,7,2,4,6,5,2,5,2,5,6,3,2,6,2,5,3,6,2,4,3,4,4,3,4,7,3
%N A327394 Number of stable divisors of n.
%C A327394 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. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476. Maximum stable divisor is A327393.
%H A327394 Antti Karttunen, <a href="/A327394/b327394.txt">Table of n, a(n) for n = 1..20000</a>
%H A327394 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 A327394 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%F A327394 a(n) = Sum_{d|n} A378442(d). - _Antti Karttunen_, Nov 27 2024
%e A327394 The stable divisors of 60 are {1, 2, 3, 4, 5, 15}, so a(60) = 6.
%t A327394 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A327394 Table[Length[Select[Divisors[n],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]],{n,100}]
%o A327394 (PARI)
%o A327394 A378442(n)={my(v=apply(primepi, factor(n)[, 1])); for(j=2, #v, for(i=1, j-1, if(v[j]%v[i]==0, return(0)))); 1}; \\ From the function "ok" in A316476 by _Andrew Howroyd_, Aug 26 2018
%o A327394 A327394(n) = sumdiv(n,d,A378442(d)); \\ _Antti Karttunen_, Nov 27 2024
%Y A327394 See link for additional cross-references.
%Y A327394 Cf. A000005, A033273, A303362, A316476, A327393.
%Y A327394 Inverse Möbius transform of A378442.
%K A327394 nonn
%O A327394 1,2
%A A327394 _Gus Wiseman_, Sep 15 2019
%E A327394 More terms from _Antti Karttunen_, Nov 27 2024