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A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

%I A340198 #24 Jan 07 2021 10:41:45
%S A340198 1,1,1,1,1,-1,1,-1,-2,-5,1,-8,1,-9,-13,-9,1,-16,1,-22,-21,-17,1,-28,
%T A340198 -14,-21,-20,-36,1,-43,1,-31,-37,-29,-45,-49,1,-33,-45,-62,1,-67,1,
%U A340198 -64,-64,-41,1,-69,-34,-64,-61,-78,1,-68,-77,-96,-69,-53,1,-88,1,-57,-96,-79,-93,-115,1,-106,-85,-123,1,-95,1,-69
%N A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.
%C A340198 Conversely, the Dirichlet inverse of this sequence yields a sequence which is one less than A319340, i.e., pointwise sum s(n) = A109606(n) + A023900(n).
%C A340198 a(9796) = 0 is the only zero among the first 2^22 terms.
%H A340198 Antti Karttunen, <a href="/A340198/b340198.txt">Table of n, a(n) for n = 1..16384</a>
%F A340198 a(1) = 1, for n > 1, a(n) = -Sum_{d|n, d<n} (A319340(n/d)-1) * a(d).
%o A340198 (PARI)
%o A340198 A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
%o A340198 A319340(n) = (eulerphi(n)+A023900(n));
%o A340198 A340198(n) = if(1==n,1,-sumdiv(n,d,if(d<n,(A319340(n/d)-1)*A340198(d),0)));
%Y A340198 Cf. A000010, A023900, A109606, A319340, A340090, A340094, A340197, A340367.
%K A340198 sign,look
%O A340198 1,9
%A A340198 _Antti Karttunen_, Jan 05 2021