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A165909
a(n) is the sum of the quadratic residues of n.
5
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, 169, 126, 84, 203, 150, 186, 72, 165, 289, 175, 96, 333, 342, 208, 135, 410, 308, 430, 198, 225, 460, 423, 124, 490, 525, 408, 299, 689, 549, 385, 252, 532, 841, 767, 270
OFFSET
1,5
COMMENTS
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.
..n..num QNRs..sum QNRs.........QNRs
..1.........0.........0
..2.........1.........1.........1
..3.........1.........1.........1
..4.........1.........1.........1
..5.........2.........5.........1..4
..6.........3.........8.........1..3..4
..7.........3.........7.........1..2..4
..8.........2.........5.........1..4
..9.........3........12.........1..4..7
.10.........5........25.........1..4..5..6..9
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
REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 4th ed., Oxford Univ. Press, 1960, pp. 88-90.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from C. H. Gribble)
MATHEMATICA
residueQ[n_, k_] := Length[Select[Range[Floor[k/2]], PowerMod[#, 2, k] == n&, 1]] == 1;
a[n_] := Select[Range[n-1], residueQ[#, n]&] // Total;
Array[a, 60] (* Jean-François Alcover, Mar 13 2019 *)
PROG
(Haskell)
import Data.List (nub)
a165909 n = sum $ nub $ map (`mod` n) $
take (fromInteger n) $ tail a000290_list
-- Reinhard Zumkeller, Aug 01 2012
(PARI) a(n) = sum(k=0, n-1, k*issquare(Mod(k, n))); \\ Michel Marcus, Mar 13 2019
CROSSREFS
Row sums of A046071 and of A096008.
Sequence in context: A204921 A021867 A346192 * A334482 A243598 A230366
KEYWORD
nonn
AUTHOR
STATUS
approved