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A347228 Dirichlet inverse of A344695, gcd(sigma(n), psi(n)).

%I A347228 #16 Aug 26 2021 16:06:26
%S A347228 1,-3,-4,8,-6,12,-8,-24,15,18,-12,-32,-14,24,24,72,-18,-45,-20,-48,32,
%T A347228 36,-24,96,35,42,-60,-64,-30,-72,-32,-216,48,54,48,120,-38,60,56,144,
%U A347228 -42,-96,-44,-96,-90,72,-48,-288,63,-105,72,-112,-54,180,72,192,80,90,-60,192,-62,96,-120,648,84,-144,-68,-144,96
%N A347228 Dirichlet inverse of A344695, gcd(sigma(n), psi(n)).
%C A347228 This is not multiplicative because A344695 is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = -484 <> -480 = 8 * -60 = a(4) * a(27).
%C A347228 Conjecture: No zeros occur as terms. Checked up to n = 2^21.
%H A347228 Antti Karttunen, <a href="/A347228/b347228.txt">Table of n, a(n) for n = 1..12000</a>
%F A347228 a(1) = 1; a(n) = -Sum_{d|n, d < n} A344695(n/d) * a(d).
%F A347228 a(n) = A347229(n) - A344695(n).
%o A347228 (PARI)
%o A347228 up_to = 16384;
%o A347228 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A347228 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
%o A347228 A344695(n) = gcd(sigma(n), A001615(n));
%o A347228 v347228 = DirInverseCorrect(vector(up_to,n,A344695(n)));
%o A347228 A347228(n) = v347228[n];
%Y A347228 Cf. A000203, A001615, A344695, A347229, A347230.
%K A347228 sign
%O A347228 1,2
%A A347228 _Antti Karttunen_, Aug 25 2021