A171555 -id:A171555 - cp's OEIS frontend

A076409 Sum of the quadratic residues of prime(n).

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

Offset: 1

R. K. Guy, Oct 08 2002

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

			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

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.

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).

Cf. A076410.

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

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

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 *)

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

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

Edited and extended by Robert G. Wilson v, Oct 09 2002

A282035 Sum of quadratic residues of (n-th prime == 3 mod 4).

Original entry on oeis.org

1, 7, 22, 76, 92, 186, 430, 423, 767, 1072, 994, 1343, 1577, 2369, 2675, 3683, 3930, 4587, 5134, 6520, 6012, 7518, 7831, 8955, 10761, 11596, 12258, 12428, 14809, 15517, 16802, 19527, 23025, 21148, 26811, 29148, 28720, 31929, 35247, 33321, 41900, 41807, 44778, 47844, 51856, 52771, 51253, 57466

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 200 do
p:=ithprime(i1);
if (p mod 4) = 3 then
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
fi;
od:
a; m; d; # A282035, A282036, A282037

Table[Table[Mod[a^2, p], {a, 1, (p-1)/2}]//Total, {p, Select[Prime[Range[100]], Mod[#, 4]==3 &]}] (* Vincenzo Librandi, Feb 21 2017 *)
Table[Total[PowerMod[#,2,n]&/@Range[n/2]],{n,Select[Prime[Range[100]],Mod[#,4]==3&]}] (* Harvey P. Dale, Dec 11 2024 *)

do(p)=sum(k=1,p-1,k^2%p)/2
apply(do, select(p->p%4==3, primes(100))) \\ Charles R Greathouse IV, Feb 21 2017

A125615 Sum of the quadratic nonresidues of prime(n).

Original entry on oeis.org

0, 2, 5, 14, 33, 39, 68, 95, 161, 203, 279, 333, 410, 473, 658, 689, 944, 915, 1139, 1491, 1314, 1738, 1826, 1958, 2328, 2525, 2884, 2996, 2943, 3164, 4318, 4585, 4658, 5004, 5513, 6191, 6123, 6683, 7849, 7439, 8413, 8145, 10314, 9264, 9653, 10746, 11394

Offset: 1

Nick Hobson, Nov 30 2006

For all n > 2, prime(n) divides a(n).

			The quadratic nonresidues of 7=prime(4) are 3, 5 and 6. Hence a(4) = 3+5+6 = 14.

D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 185.

Nick Hobson, 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.

Cf. A076409, A076410, A125613-A125618.

Sums of residues, nonresidues, and their differences, for p == 1 (mod 4), p == 3 (mod 4), and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Table[p=Prime[n];Total[Complement[Range[p-1],Union[Table[PowerMod[k, 2, p], {k, p}]]]],{n,47}] (* James C. McMahon, Dec 19 2024 *)

vector(47, n, p=prime(n); t=1; for(i=2, (p-1)/2, t+=((i^2)%p)); p*(p-1)/2-t)

If prime(n) = 4k+1 then a(n) = k(4k+1) = A076409(n).

A282036 a(n) is the sum of quadratic nonresidues of A002145(n) (the n-th prime == 3 mod 4).

Original entry on oeis.org

2, 14, 33, 95, 161, 279, 473, 658, 944, 1139, 1491, 1738, 1826, 2884, 2996, 4318, 4585, 5004, 6191, 6683, 7849, 8413, 10314, 10746, 11394, 13157, 13393, 16013, 16566, 18936, 19783, 20376, 23946, 27057, 27804, 30883, 35541, 35232, 36384, 39832, 45671, 50858, 51363, 50059, 55097, 56040

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Christian Aebi, Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Cf. A002145.

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 200 do
p:=ithprime(i1);
if (p mod 4) = 3 then
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
fi;
od:
a; m; d; # A282035, A282036, A282037
# Alternative:
f:= p -> add(-k^2 mod p, k=1..(p-1)/2)::
map(f, select(isprime, [seq(p,p=3..1000,4)])); # Robert Israel, Nov 09 2020

f[p_] := Total[Range[p-1] ~Complement~ Table[Mod[k^2, p], {k, (p-1)/2}] ]; f /@ Select[Range[3, 1000, 4], PrimeQ] (* Jean-François Alcover, Feb 16 2018, after Robert Israel *)

lista(nn) = forprime(p=2, nn, if(p%4==3, print1(sum(k=1, p-1, if (!issquare(Mod(k, p)), k)), ", "))); \\ Michel Marcus, Nov 09 2020

a(n) = Sum_{k=1..(A002145(n)-1)/2} (-k^2) mod A002145(n). - J. M. Bergot and Robert Israel, Nov 09 2020

A282037 Let p = n-th prime == 3 mod 4; a(n) = (sum of quadratic nonresidues mod p) - (sum of quadratic residues mod p).

Original entry on oeis.org

1, 7, 11, 19, 69, 93, 43, 235, 177, 67, 497, 395, 249, 515, 321, 635, 655, 417, 1057, 163, 1837, 895, 2483, 1791, 633, 1561, 1135, 3585, 1757, 3419, 2981, 849, 921, 5909, 993, 1735, 6821, 3303, 1137, 6511, 3771, 9051, 6585, 2215, 3241, 3269, 11975, 3409, 4419, 1497, 10563, 2615, 1641, 5067, 2855

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Equals A282036 - A282035.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Christian Aebi and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT] (2015).

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 200 do
p:=ithprime(i1);
if (p mod 4) = 3 then
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
fi;
od:
a; m; d; # A282035, A282036, A282037

sum[p_] := Total[If[JacobiSymbol[#, p] == 1, -#, #]& /@ Range[p-1]];
sum /@ Select[Prime[Range[200]], Mod[#, 4] == 3&] (* Jean-François Alcover, Aug 31 2018 *)

A282038 (Sum of the quadratic nonresidues of prime(n)) - (sum of the quadratic residues of prime(n)).

Original entry on oeis.org

-1, 1, 0, 7, 11, 0, 0, 19, 69, 0, 93, 0, 0, 43, 235, 0, 177, 0, 67, 497, 0, 395, 249, 0, 0, 0, 515, 321, 0, 0, 635, 655, 0, 417, 0, 1057, 0, 163, 1837, 0, 895, 0, 2483, 0, 0, 1791, 633, 1561, 1135, 0, 0, 3585, 0, 1757, 0, 3419, 0, 2981, 0, 0, 849, 0, 921, 5909, 0, 0, 993, 0, 1735, 0, 0, 6821, 3303, 0

Offset: 1

N. J. A. Sloane, Feb 20 2017

sign

Equals 0 if p == 1 (mod 4).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT] (2015).

Sums of residues, nonresidues, and their differences, for p == 1 (mod 4), p == 3 (mod 4), and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 100 do
p:=ithprime(i1);
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
od:
a; m; d; # A076409, A125615, A282038

sum[p_] := Total[If[JacobiSymbol[#, p] == 1, -#, #]& /@ Range[p-1]];
a[n_] := sum[Prime[n]];
Array[a, 100] (* Jean-François Alcover, Aug 31 2018 *)

a(n) = my (p=prime(n)); return (sum(i=1, p-1, if (kronecker(i,p)==1, -i, +i))) \\ Rémy Sigrist, Apr 28 2017

cp's OEIS Frontend

A076409 Sum of the quadratic residues of prime(n).

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A282035 Sum of quadratic residues of (n-th prime == 3 mod 4).

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A125615 Sum of the quadratic nonresidues of prime(n).

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A282036 a(n) is the sum of quadratic nonresidues of A002145(n) (the n-th prime == 3 mod 4).

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A282037 Let p = n-th prime == 3 mod 4; a(n) = (sum of quadratic nonresidues mod p) - (sum of quadratic residues mod p).

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A282038 (Sum of the quadratic nonresidues of prime(n)) - (sum of the quadratic residues of prime(n)).

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Maple

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