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A359577 Dirichlet inverse of A085731, where A085731 is the greatest common divisor of n and the arithmetic derivative of n.

%I A359577 #15 Jan 08 2023 15:46:59
%S A359577 1,-1,-1,-3,-1,1,-1,3,-2,1,-1,3,-1,1,1,-3,-1,2,-1,3,1,1,-1,-3,-4,1,
%T A359577 -22,3,-1,-1,-1,3,1,1,1,6,-1,1,1,-3,-1,-1,-1,3,2,1,-1,3,-6,4,1,3,-1,
%U A359577 22,1,-3,1,1,-1,-3,-1,1,2,-3,1,-1,-1,3,1,-1,-1,-6,-1,1,4,3,1,-1,-1,3,28,1,-1,-3,1,1,1,-3,-1,-2,1,3,1,1,1,-3,-1,6,2,12,-1,-1,-1,-3,-1
%N A359577 Dirichlet inverse of A085731, where A085731 is the greatest common divisor of n and the arithmetic derivative of n.
%C A359577 Multiplicative because A085731 is.
%H A359577 Antti Karttunen, <a href="/A359577/b359577.txt">Table of n, a(n) for n = 1..65537</a>
%F A359577 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A085731(n/d) * a(d).
%p A359577 g:= proc(n) option remember;
%p A359577       igcd(n, n*add(i[2]/i[1], i=ifactors(n)[2]))
%p A359577     end:
%p A359577 a:= proc(n) option remember; `if`(n=1, 1, -add(
%p A359577       a(d)*g(n/d), d=numtheory[divisors](n) minus {n}))
%p A359577     end:
%p A359577 seq(a(n), n=1..120);  # _Alois P. Heinz_, Jan 07 2023
%t A359577 d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); s[n_] := GCD[n, d[n]]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Jan 07 2023 *)
%Y A359577 Cf. A003415, A085731, A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).
%K A359577 sign,mult
%O A359577 1,4
%A A359577 _Antti Karttunen_, Jan 06 2023