OFFSET
1,3
COMMENTS
Row sums of A063987. - R. J. Mathar, Jan 08 2015
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
REFERENCES
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.
LINKS
Zak Seidov, Table of n, a(n) for n = 1..1000
Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT] (2015).
FORMULA
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
EXAMPLE
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
MAPLE
A076409 := proc(n)
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 ;
end proc: # R. J. Mathar, Feb 26 2011
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. K. Guy, Oct 08 2002
EXTENSIONS
Edited and extended by Robert G. Wilson v, Oct 09 2002
STATUS
approved