cp's OEIS Frontend

A340188 Sum of A063994 and its Dirichlet inverse, where A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).

%I A340188 #7 Dec 31 2020 15:34:52
%S A340188 2,0,0,1,0,4,0,1,4,8,0,0,0,12,16,1,0,-4,0,-2,24,20,0,1,16,24,0,-4,0,
%T A340188 -28,0,1,40,32,48,5,0,36,48,1,0,-48,0,-8,-16,44,0,1,36,-32,64,-10,0,8,
%U A340188 80,5,72,56,0,24,0,60,-32,1,96,-88,0,-14,88,-116,0,0,0,72,-48,-16,120,-108,0,1,4,80,0,48,128,84,112
%N A340188 Sum of A063994 and its Dirichlet inverse, where A063994(x) = Product_{primes p dividing x} gcd(p-1, x-1).
%H A340188 Antti Karttunen, <a href="/A340188/b340188.txt">Table of n, a(n) for n = 1..8191</a>
%H A340188 Antti Karttunen, <a href="/A340188/a340188.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A340188 a(n) = A063994(n) + A340187(n).
%F A340188 a(n) = A340189(n) - A318828(n).
%o A340188 (PARI)
%o A340188 up_to = 65537;
%o A340188 A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
%o A340188 DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(d<n, v[n/d]*u[d], 0))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A340188 v340187 = DirInverse(vector(up_to, n, A063994(n)));
%o A340188 A340187(n) = v340187[n];
%o A340188 A340188(n) = (A063994(n)+A340187(n));
%Y A340188 Cf. A063994, A318828, A340187, A340189.
%Y A340188 Cf. also A319340, A340191.
%K A340188 sign
%O A340188 1,1
%A A340188 _Antti Karttunen_, Dec 31 2020