A092509 Möbius transform of sequence A008475.

0, 2, 3, 2, 5, 0, 7, 4, 6, 0, 11, 0, 13, 0, 0, 8, 17, 0, 19, 0, 0, 0, 23, 0, 20, 0, 18, 0, 29, 0, 31, 16, 0, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 0, 0, 47, 0, 42, 0, 0, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 0, 32, 0, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 54, 0, 83, 0, 0, 0, 0, 0, 89, 0, 0, 0, 0

Offset: 1

Author

Leroy Quet, Dec 31 2004

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A008475, A008683.

Programs

Mathematica

f[n_] := Plus @@ Map[Power @@ #1 &, FactorInteger[n]]; mt[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu /@ (n/d)*f /@ d)]; Table[ mt[n], {n, 93}] (* Robert G. Wilson v, Jan 12 2005 *)

PARI

A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); };
A092509(n) = sumdiv(n,d,moebius(d)*A008475(n/d)); \\ Antti Karttunen, Nov 17 2017

Formula

a(n) = 0 if n is 1 or not a power of a prime;

a(n) = n if n is a prime;

a(n) = n*(1 -1/p) if n = p^k, k>= 2, p=prime.

a(n) = Sum_{d|n} A008683(d)*A008475(n/d). - Antti Karttunen, Nov 17 2017

Extensions

More terms from Robert G. Wilson v, Jan 13 2005