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%I #7 Feb 17 2023 07:38:49
%S 0,-1,-4,-1,-6,-4,6,-1,8,-6,-12,2,-14,6,12,-1,16,11,-20,-6,-12,-12,22,
%T 2,24,-14,-16,6,-30,22,30,-1,24,16,-24,11,-38,-20,28,4,40,-12,-44,-12,
%U -14,22,46,2,48,29,-32,-14,-54,-16,48,-8,40,-30,-60,22,-62,30,46,-1,56,24,-68,16,-44,-38,70,11,72,-38,-28,-20,-96,28
%N Sum_{d|n, sqrt(n) < d <= n} Jacobi(2,d)*d - Sum_{d|n, 1 <= d < sqrt(n)} Jacobi(2,d)*d..
%D H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 323.
%p with(numtheory); A117001:=proc(n) local d,t1,t2; t1:=0; t2:=0; for d from 1 to n do if n mod d = 0 then if d^2>n then t1:=t1+jacobi(2,d)*d; fi; if d^2<n then t2:=t2+jacobi(2,d)*d; fi; fi; od: t1-t2; end;
%t a[n_] := Sum[Which[Sqrt[n]<d<=n, 1, 1<=d<Sqrt[n], -1, True, 0]*JacobiSymbol[2, d]*d, {d, Divisors[n]}];
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 17 2023 *)
%Y Cf. A117000.
%K sign
%O 1,3
%A _N. J. A. Sloane_, Apr 15 2006
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