This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379219 #10 Jan 03 2025 05:17:51
%S A379219 1,-1,-2,-1,-5,2,-6,-1,-3,5,-11,2,-13,6,10,-1,-17,3,-19,5,12,11,-23,2,
%T A379219 0,13,0,6,-29,-10,-30,-1,22,17,30,3,-37,19,26,5,-41,-12,-43,11,15,23,
%U A379219 -47,2,-7,0,34,13,-53,0,55,6,38,29,-59,-10,-61,30,18,-1,65,-22,-67,17,46,-30,-71,3,-73,37,0,19,66,-26,-79,5
%N A379219 Dirichlet inverse of A379218.
%H A379219 Antti Karttunen, <a href="/A379219/b379219.txt">Table of n, a(n) for n = 1..32769</a>
%H A379219 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A379219 a(n) = Sum_{d|n} A379109(d).
%F A379219 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A379218(n/d) * a(d).
%F A379219 Multiplicative with a(2^e) = -1, and for an odd prime p, if p is a Mersenne prime, a(p) = 1-p, a(p^2) = -p, and a(p^e) = 0 for e >= 3, and otherwise a(p) = -p and a(p^e) = 0 for e >= 2. - _Amiram Eldar_, Jan 03 2025
%t A379219 f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, Which[e == 1, 1 - p, e == 2, -p, e > 2, 0], If[e == 1, -p, 0]]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 03 2025 *)
%o A379219 (PARI)
%o A379219 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 A379219 A209229(n) = (n && !bitand(n,n-1));
%o A379219 A336923(n) = A209229(sigma(n+n)-sigma(n));
%o A379219 A379109(n) = sumdiv(n,d,A046692(d)*A336923(n/d));
%o A379219 A379219(n) = sumdiv(n,d,A379109(d));
%Y A379219 Inverse Möbius transform of A379109.
%Y A379219 Dirichlet inverse of A379218.
%Y A379219 Cf. A000668, A046692, A336923.
%K A379219 sign,mult,easy
%O A379219 1,3
%A A379219 _Antti Karttunen_, Dec 18 2024