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A349911 Dirichlet inverse of A336466, which is fully multiplicative with a(p) = oddpart(p-1).

%I A349911 #15 Dec 10 2021 22:56:06
%S A349911 1,-1,-1,0,-1,1,-3,0,0,1,-5,0,-3,3,1,0,-1,0,-9,0,3,5,-11,0,0,3,0,0,-7,
%T A349911 -1,-15,0,5,1,3,0,-9,9,3,0,-5,-3,-21,0,0,11,-23,0,0,0,1,0,-13,0,5,0,9,
%U A349911 7,-29,0,-15,15,0,0,3,-5,-33,0,11,-3,-35,0,-9,9,0,0,15,-3,-39,0,0,5,-41,0,1,21,7,0,-11,0
%N A349911 Dirichlet inverse of A336466, which is fully multiplicative with a(p) = oddpart(p-1).
%C A349911 Multiplicative because A336466 is.
%H A349911 Antti Karttunen, <a href="/A349911/b349911.txt">Table of n, a(n) for n = 1..16384</a>
%F A349911 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336466(n/d) * a(d).
%F A349911 a(n) = A349912(n) - A336466(n).
%t A349911 f[p_, e_] := ((p-1)/2^IntegerExponent[p-1, 2])^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Dec 08 2021 *)
%o A349911 (PARI)
%o A349911 A000265(n) = (n>>valuation(n,2));
%o A349911 A336466(n) = { my(f=factor(n)); prod(k=1,#f~,A000265(f[k,1]-1)^f[k,2]); };
%o A349911 memoA349911 = Map();
%o A349911 A349911(n) = if(1==n,1,my(v); if(mapisdefined(memoA349911,n,&v), v, v = -sumdiv(n,d,if(d<n,A336466(n/d)*A349911(d),0)); mapput(memoA349911,n,v); (v)));
%Y A349911 Cf. A336466, A349912.
%Y A349911 Cf. also A097945.
%K A349911 sign,mult
%O A349911 1,7
%A A349911 _Antti Karttunen_, Dec 08 2021