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A076409
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Sum of the quadratic residues of prime(n).
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18
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1, 1, 5, 7, 22, 39, 68, 76, 92, 203, 186, 333, 410, 430, 423, 689, 767, 915, 1072, 994, 1314, 1343, 1577, 1958, 2328, 2525, 2369, 2675, 2943, 3164, 3683, 3930, 4658, 4587, 5513, 5134, 6123, 6520, 6012, 7439, 7518, 8145, 7831, 9264, 9653, 8955, 10761, 11596
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OFFSET
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1,3
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COMMENTS
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prime(n) divides a(n) for n > 2. This is implied by a variant of Wolstenholme's theorem (see Hardy & Wright reference). - Isaac Saffold, Jun 21 2018
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 4th ed., Oxford Univ. Press, 1960, p. 88-90.
Kenneth A. Ribet, Modular forms and Diophantine questions, Challenges for the 21st century (Singapore 2000), 162-182; World Sci. Publishing, River Edge NJ 2001; Math. Rev. 2002i:11030.
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LINKS
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FORMULA
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If prime(n) = 4k+1 then a(n) = k*(4k+1).
For n>2 if prime(n) = 4k+3 then a(n) = (k - b)*(4k+3) where b = (h(-p) - 1) / 2; h(-p) = A002143. For instance. If n=5, p=11, k=2, b=(1-1)/2=0 and a(5) = 2*11 = 22. If n=20, p=71, k=17, b=(7-1)/2=3 and a(20) = 14*71 = 994. - Andrés Ventas, Mar 01 2021
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EXAMPLE
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If n = 3, then p = 5 and a(3) = 1 + 4 = 5. If n = 4, then p = 7 and a(4) = 1 + 4 + 2 = 7. If n = 5, then p = 11 and a(5) = 1 + 4 + 9 + 5 + 3 = 22. - Michael Somos, Jul 01 2018
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MAPLE
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local a, p, i ;
p := ithprime(n) ;
a := 0 ;
for i from 1 to p-1 do
if numtheory[legendre](i, p) = 1 then
a := a+i ;
end if;
end do;
a ;
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MATHEMATICA
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Join[{1, 1}, Table[ Apply[ Plus, Flatten[ Position[ Table[ JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]]], {n, 3, 48}]]
Join[{1}, Table[p=Prime[n]; If[Mod[p, 4]==1, p(p-1)/4, Sum[PowerMod[k, 2, p], {k, p/2}]], {n, 2, 1000}]] (* Zak Seidov, Nov 02 2011 *)
a[ n_] := If[ n < 3, Boole[n > 0], With[{p = Prime[n]}, Sum[ Mod[k^2, p], {k, (p - 1)/2}]]]; (* Michael Somos, Jul 01 2018 *)
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PROG
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(PARI) a(n, p=prime(n))=if(p<5, return(1)); if(k%4==1, return(p\4*p)); sum(k=1, p-1, k^2%p) \\ Charles R Greathouse IV, Feb 21 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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