A327394 Number of stable divisors of n.

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, 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, 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

Offset: 1

Author

Gus Wiseman, Sep 15 2019

Keywords

• nonn

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. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476. Maximum stable divisor is A327393.

Examples

			The stable divisors of 60 are {1, 2, 3, 4, 5, 15}, so a(60) = 6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Gus Wiseman, Sequences counting and encoding certain classes of multisets
• Index entries for sequences related to prime indices in the factorization of n

Crossrefs

See link for additional cross-references.

Cf. A000005, A033273, A303362, A316476, A327393.

Inverse Möbius transform of A378442.

Programs

Mathematica

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Divisors[n],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]],{n,100}]

PARI

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
A327394(n) = sumdiv(n,d,A378442(d)); \\ Antti Karttunen, Nov 27 2024

Formula

a(n) = Sum_{d|n} A378442(d). - Antti Karttunen, Nov 27 2024

Extensions

More terms from Antti Karttunen, Nov 27 2024