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A229344
Mertens's function of the arithmetic derivative of n: a(n) = M(n'), a(1) = 0.
3
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
1,6
COMMENTS
a(1) = 0 by convention. - Antti Karttunen, Nov 01 2017
LINKS
FORMULA
a(1) = 0 and for n > 1, a(n) = A002321(A003415(n)).
EXAMPLE
For n=4, M(n') = M(4') = M(4) = -1.
For n=7, M(n') = M(7') = M(1) = 1.
MATHEMATICA
Array[Total@ Map[MoebiusMu, Range@ If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]]] &, 89] (* Michael De Vlieger, Nov 01 2017 *)
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Luca Brigada Villa, Sep 24 2013
EXTENSIONS
Description and formula clarified and more terms added by Antti Karttunen, Nov 01 2017
STATUS
approved