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A379109 Dirichlet convolution of A046692 (inverse of sigma) with A336923, where A336923(n) = 1 if sigma(2n) - sigma(n) is a power of 2, otherwise 0.

%I A379109 #13 Jan 03 2025 05:19:51
%S A379109 1,-2,-3,0,-6,6,-7,0,-1,12,-12,0,-14,14,18,0,-18,2,-20,0,21,24,-24,0,
%T A379109 5,28,3,0,-30,-36,-31,0,36,36,42,0,-38,40,42,0,-42,-42,-44,0,6,48,-48,
%U A379109 0,-1,-10,54,0,-54,-6,72,0,60,60,-60,0,-62,62,7,0,84,-72,-68,0,72,-84,-72,0,-74,76,-15,0,84,-84,-80,0,0,84
%N A379109 Dirichlet convolution of A046692 (inverse of sigma) with A336923, where A336923(n) = 1 if sigma(2n) - sigma(n) is a power of 2, otherwise 0.
%H A379109 Antti Karttunen, <a href="/A379109/b379109.txt">Table of n, a(n) for n = 1..32769</a>
%H A379109 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A379109 a(n) = Sum_{d|n} A046692(d)*A336923(n/d).
%F A379109 Multiplicative with a(2^e) = -2 if e = 1 and 0 otherwise, and for an odd prime p, if p is a Mersenne prime, a(p) = -p, a(p^2) = -1, a(p^3) = p, and a(p^e) = 0 for e >= 4, and otherwise a(p) = -(p+1), a(p^2) = p and a(p^e) = 0 for e >= 3. - _Amiram Eldar_, Jan 03 2025
%t A379109 f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, Which[e == 1, -p, e == 2, -1, e == 3, p, e > 3, 0], Which[e == 1, -p - 1, e == 2, p, e > 2, 0]]; f[2, e_] := If[e == 1, -2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 03 2025 *)
%o A379109 (PARI)
%o A379109 A046692(n) = { my(f=factor(n)~); prod(i=1, #f, if(1==f[2,i], -(f[1,i]+1), if(2==f[2,i], f[1,i], 0))); };
%o A379109 A209229(n) = (n && !bitand(n,n-1));
%o A379109 A336923(n) = A209229(sigma(n+n)-sigma(n));
%o A379109 A379109(n) = sumdiv(n,d,A046692(d)*A336923(n/d));
%Y A379109 Cf. A000203, A000668, A046692, A054784, A336923, A379108 (Dirichlet inverse).
%K A379109 sign,mult,easy
%O A379109 1,2
%A A379109 _Antti Karttunen_, Dec 17 2024