A229344 Mertens's function of the arithmetic derivative of n: a(n) = M(n'), a(1) = 0.

0, 1, 1, -1, 1, -2, 1, -2, -1, -2, 1, -1, 1, -2, -2, -4, 1, -2, 1, -2, -1, -3, 1, -3, -1, -1, -1, -4, 1, -4, 1, -4, -2, -3, -2, -1, 1, -2, -1, -2, 1, -1, 1, -3, 0, -2, 1, -4, -2, -3, -3, -2, 1, -4, -1, -1, -1, -4, 1, -1, 1, -3, -2, -5, -2, -2, 1, -3, -1, -1, 1, -1, 1, 0, -2, -4, -2, -3, 1, -4, -3, -3, 1, -1, -1, -3, -4, -4, 1

Offset: 1

Author

Luca Brigada Villa, Sep 24 2013

Keywords

• sign

Comments

a(1) = 0 by convention. - Antti Karttunen, Nov 01 2017

Examples

			For n=4, M(n') = M(4') = M(4) = -1.
For n=7, M(n') = M(7') = M(1) = 1.
		

Links

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

Crossrefs

Cf. A002321, A003415, A229340, A229341, A229342, A229343.

Programs

Mathematica

Array[Total@ Map[MoebiusMu, Range@ If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]]] &, 89] (* Michael De Vlieger, Nov 01 2017 *)

PARI

rd(n) = {local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1];));}
a(n) = mertens(rd(n)); \\ Michel Marcus, Sep 24 2013

PARI

A002321(n) = sum(k=1, n, moebius(k));
A003415(n) = { my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From Michael B. Porter, Nov 25 2009
A229344(n) = if(1==n,0,A002321(A003415(n))); \\ Antti Karttunen, Nov 01 2017

Formula

a(1) = 0 and for n > 1, a(n) = A002321(A003415(n)).

Extensions

Description and formula clarified and more terms added by Antti Karttunen, Nov 01 2017