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A349912 Sum of A336466 and its Dirichlet inverse, where A336466 is fully multiplicative with a(p) = oddpart(p-1).

%I A349912 #13 Dec 10 2021 22:55:57
%S A349912 2,0,0,1,0,2,0,1,1,2,0,1,0,6,2,1,0,1,0,1,6,10,0,1,1,6,1,3,0,0,0,1,10,
%T A349912 2,6,1,0,18,6,1,0,0,0,5,1,22,0,1,9,1,2,3,0,1,10,3,18,14,0,1,0,30,3,1,
%U A349912 6,0,0,1,22,0,0,1,0,18,1,9,30,0,0,1,1,10,0,3,2,42,14,5,0,1,18,11,30,46,18,1,0,9,5,1
%N A349912 Sum of A336466 and its Dirichlet inverse, where A336466 is fully multiplicative with a(p) = oddpart(p-1).
%H A349912 Antti Karttunen, <a href="/A349912/b349912.txt">Table of n, a(n) for n = 1..16384</a>
%F A349912 a(n) = A336466(n) + A349911(n).
%F A349912 a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A336466(d) * A349911(n/d).
%F A349912 a(4*n) = A336466(n).
%t A349912 f[p_, e_] := ((p-1)/2^IntegerExponent[p-1, 2])^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := s[n] + sinv[n]; Array[a, 100] (* _Amiram Eldar_, Dec 08 2021 *)
%o A349912 (PARI)
%o A349912 A000265(n) = (n>>valuation(n,2));
%o A349912 A336466(n) = { my(f=factor(n)); prod(k=1,#f~,A000265(f[k,1]-1)^f[k,2]); };
%o A349912 memoA349911 = Map();
%o A349912 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)));
%o A349912 A349912(n) = (A336466(n)+A349911(n));
%Y A349912 Cf. A336466 (also a quadrisection of this sequence), A349911.
%Y A349912 Cf. also A322581.
%K A349912 nonn
%O A349912 1,1
%A A349912 _Antti Karttunen_, Dec 08 2021