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A165909 a(n) is the sum of the quadratic residues of n.

%I A165909 #54 Feb 28 2021 09:53:47
%S A165909 0,1,1,1,5,8,7,5,12,25,22,14,39,42,30,14,68,60,76,35,70,110,92,42,125,
%T A165909 169,126,84,203,150,186,72,165,289,175,96,333,342,208,135,410,308,430,
%U A165909 198,225,460,423,124,490,525,408,299,689,549,385,252,532,841,767,270
%N A165909 a(n) is the sum of the quadratic residues of n.
%C A165909 The table below shows n, the number of nonzero quadratic residues (QRs) of n (A105612), the sum of the QRs of n and the nonzero QRs of n (A046071) for n = 1..10.
%C A165909 ..n..num QNRs..sum QNRs.........QNRs
%C A165909 ..1.........0.........0
%C A165909 ..2.........1.........1.........1
%C A165909 ..3.........1.........1.........1
%C A165909 ..4.........1.........1.........1
%C A165909 ..5.........2.........5.........1..4
%C A165909 ..6.........3.........8.........1..3..4
%C A165909 ..7.........3.........7.........1..2..4
%C A165909 ..8.........2.........5.........1..4
%C A165909 ..9.........3........12.........1..4..7
%C A165909 .10.........5........25.........1..4..5..6..9
%C A165909 When p is prime >= 5, a(p) is a multiple of p by a variant of Wolstenholme's theorem (see A076409 and A076410). _Robert Israel_ remarks that we don't need Wolstenholme, just the fact that Sum_{x=1..p-1} x^2 = p*(2*p-1)*(p-1)/6. - _Bernard Schott_, Mar 13 2019
%D A165909 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 4th ed., Oxford Univ. Press, 1960, pp. 88-90.
%H A165909 Alois P. Heinz, <a href="/A165909/b165909.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from C. H. Gribble)
%t A165909 residueQ[n_, k_] := Length[Select[Range[Floor[k/2]], PowerMod[#, 2, k] == n&, 1]] == 1;
%t A165909 a[n_] := Select[Range[n-1], residueQ[#, n]&] // Total;
%t A165909 Array[a, 60] (* _Jean-François Alcover_, Mar 13 2019 *)
%o A165909 (Haskell)
%o A165909 import Data.List (nub)
%o A165909 a165909 n = sum $ nub $ map (`mod` n) $
%o A165909                         take (fromInteger n) $ tail a000290_list
%o A165909 -- _Reinhard Zumkeller_, Aug 01 2012
%o A165909 (PARI) a(n) = sum(k=0, n-1, k*issquare(Mod(k,n))); \\ _Michel Marcus_, Mar 13 2019
%Y A165909 Row sums of A046071 and of A096008.
%Y A165909 Cf. A000290, A076409, A076410.
%K A165909 nonn
%O A165909 1,5
%A A165909 _Christopher Hunt Gribble_, Sep 30 2009