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A083344 a(n) = A082457(n) - A066715(n) = gcd(2n+1, A057643(2n+1)) - gcd(2n+1, A000203(2n+1)).

%I A083344 #14 Nov 17 2017 04:14:14
%S A083344 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,
%T A083344 0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,
%U A083344 0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,30,0,0,0,0,0,0,4,0,0
%N A083344 a(n) = A082457(n) - A066715(n) = gcd(2n+1, A057643(2n+1)) - gcd(2n+1, A000203(2n+1)).
%H A083344 Antti Karttunen, <a href="/A083344/b083344.txt">Table of n, a(n) for n = 1..32768</a>
%F A083344 a(n) = gcd(2n+1, lcm(1+D(2n+1))) - gcd(2n+1, sigma(2n+1)), gcd(2n+1, A057643(2n+1)) - gcd(2n+1, A000203(2n+1)), where D(x) is the set of divisors of x.
%e A083344 n=22: 2n+1 = 45, A057643(45) = 5520, a(22) = gcd(45,5520) = 15 while A066715(45) = 3; a(22) = 15-3 = 12; sites where nonzero terms appear see in A082452.
%t A083344 di[x_] := Apply[LCM, Divisors[x]+1] (*A066715=*)t1=Table[GCD[2*n+1, DivisorSigma[1, 2*n+1]], {n, 1, 2048}]; (*A082457=*)t2=Table[GCD[2*w+1, di[1+2*w]], {w, 1, 2048}]; (*A083344=*)t2-t1;
%o A083344 (PARI) a(n)=gcd(lcm(apply(d->d+1,divisors(2*n+1))),2*n+1)-gcd(sigma(2*n+1),2*n+1) \\ _Charles R Greathouse IV_, Feb 14 2013
%Y A083344 Cf. A057643, A066715, A000203, A082457, A082452.
%K A083344 sign
%O A083344 1,22
%A A083344 _Labos Elemer_, Apr 25 2003