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A349340 Dirichlet inverse of A003557, where A003557 is multiplicative with a(p^e) = p^(e-1).

%I A349340 #13 Nov 25 2021 18:53:53
%S A349340 1,-1,-1,-1,-1,1,-1,-1,-2,1,-1,1,-1,1,1,-1,-1,2,-1,1,1,1,-1,1,-4,1,-4,
%T A349340 1,-1,-1,-1,-1,1,1,1,2,-1,1,1,1,-1,-1,-1,1,2,1,-1,1,-6,4,1,1,-1,4,1,1,
%U A349340 1,1,-1,-1,-1,1,2,-1,1,-1,-1,1,1,-1,-1,2,-1,1,4,1,1,-1,-1,1,-8,1,-1,-1,1,1,1,1,-1,-2,1
%N A349340 Dirichlet inverse of A003557, where A003557 is multiplicative with a(p^e) = p^(e-1).
%H A349340 Antti Karttunen, <a href="/A349340/b349340.txt">Table of n, a(n) for n = 1..20000</a>
%F A349340 Multiplicative with a(p^e) = -((p-1)^(e-1)).
%F A349340 a(n) = A076479(n) * A326297(n).
%F A349340 a(1) = 1; a(n) = -Sum_{d|n, d < n} A003557(n/d) * a(d).
%t A349340 f[p_, e_] := -(p - 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349340 (PARI)
%o A349340 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A349340 memoA349340 = Map();
%o A349340 A349340(n) = if(1==n,1,my(v); if(mapisdefined(memoA349340,n,&v), v, v = -sumdiv(n,d,if(d<n,A003557(n/d)*A349340(d),0)); mapput(memoA349340,n,v); (v)));
%o A349340 (PARI) A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };
%Y A349340 Cf. A003557, A076479, A326297 (absolute values).
%Y A349340 Cf. also A325126, A349350, A349619.
%K A349340 sign,mult
%O A349340 1,9
%A A349340 _Antti Karttunen_, Nov 18 2021