A282721 -id:A282721 - cp's OEIS frontend

A282043 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic nonresidues mod p.

14, 161, 279, 658, 1491, 1738, 2884, 4318, 6191, 7849, 10314, 10746, 13157, 16013, 18936, 19783, 27057, 35541, 35232, 39832, 50858, 51363, 55097, 63228, 60875, 68408, 97038, 95906, 103484, 111931, 140205, 136676, 145628, 146445, 172830, 189614, 195038, 209332, 221373, 219641, 238849, 254597

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282042 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 7 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282039, A282040, A282041, A282039 again, A282042, A282043

sqnr[p_] := Select[Range[p-1], JacobiSymbol[#, p] != 1&] // Total;
sqnr /@ Select[Prime[Range[200]], Mod[#, 8] == 7&] (* Jean-François Alcover, Aug 30 2018 *)

A282727 Let p = n-th prime == 3 mod 8; a(n) = (sum of quadratic residues mod p that are < p/2) + (sum of all quadratic residues mod p).

Original entry on oeis.org

2, 35, 108, 567, 1073, 1386, 2132, 3551, 5330, 6003, 8262, 9968, 13860, 16046, 19625, 24957, 29376, 34155, 37541, 44793, 54758, 61217, 68036, 75215, 77688, 85347, 93366, 98912, 101745, 107531, 119583, 129042, 135548, 145607, 149040, 170478, 193356, 205335, 213521, 230373, 243432, 256851, 280016, 294395

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

This is also the (sum of quadratic nonresidues mod p that are < p/2) + (sum of all quadratic nonresidues mod p) (= A282721 + A282723 = A282724 + A282726).

Robert Israel, Table of n, a(n) for n = 1..4000
Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Cf. A282035-A282043 and A282721-A282726.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282721 - A282727
# Alternative:
v:= proc(x,r) if x <= r then 2*x else x fi end proc:
f:= proc(p) local q, r, j;
  r:= (p-1)/2;
  add(v(j^2 mod p, r), j=1..r)
end proc:
map(f, select(isprime, [seq(i,i=3..1000,8)])); # Robert Israel, Mar 27 2017

v[x_, r_] := If[x <= r, 2*x, x];
f[p_] := Module[{r}, r = (p-1)/2; Sum[v[PowerMod[j, 2, p], r], {j, 1, r}]];
f /@ Select[Range[3, 1000, 8], PrimeQ] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

from sympy import isprime
def v(x, r):
    return 2*x if x<=r else x
def a(p):
    r=(p - 1)//2
    return sum(v((j**2)%p, r) for j in range(1, r + 1))
print([a(p) for p in range(3, 2001, 8) if isprime(p)]) # Indranil Ghosh, Mar 27 2017 translated from  Robert Israel's Maple program

A282039 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p that are < p/2.

Original entry on oeis.org

3, 33, 60, 138, 315, 390, 663, 1008, 1425, 1743, 2280, 2475, 3108, 3570, 4323, 4590, 6045, 8055, 8418, 9168, 11610, 12045, 13398, 14340, 14823, 15813, 22425, 23028, 24885, 26163, 32310, 33033, 34503, 35250, 42333, 43995, 46548, 49173, 51870, 52785, 58443, 60393, 61380, 66435, 67470, 70623

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

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

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 7 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
  q:=q+j;
  if jA282039, A282040, A282041, A282039 again, A282042, A282043
# alternative:
g:= proc(t,p) if t < p/2 then t else 0 fi end proc;
f:= proc(n) local k;
     add(g(k^2 mod n, n),k=1..n/2)
end proc:
P:= select(isprime, [seq(i,i=7..3000,8)]):
map(f,P); # Robert Israel, Nov 09 2020

sum[p_]:= Total[If[#Jean-François Alcover, Aug 31 2018 *)

A282040 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p that are > p/2.

Original entry on oeis.org

4, 59, 126, 285, 679, 953, 1706, 2675, 3709, 4269, 5551, 6480, 8488, 8858, 11194, 12212, 15103, 20665, 23511, 24153, 30197, 32733, 38458, 36913, 42643, 42032, 59638, 64987, 70396, 70887, 85606, 94192, 95522, 99930, 123090, 117932, 130367, 134436, 141262, 149395, 169769, 167663, 175469

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Robert Israel, 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.

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 7 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282039, A282040, A282041, A282039 again, A282042, A282043
# alternative:
g:= proc(t, p) if t > p/2 then t else 0 fi end proc;
f:= proc(n) local k;
     add(g(k^2 mod n, n), k=1..n/2)
end proc:
P:= select(isprime, [seq(i, i=7..3000, 8)]):
map(f, P); # Robert Israel, Aug 22 2025

sum[p_]:= Total[If[#>p/2 && JacobiSymbol[#, p] == 1, #, 0]& /@ Range[p-1]];
sum /@ Select[Range[7, 1000, 8], PrimeQ] (* Jean-François Alcover, Aug 31 2018 *)

A282041 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p.

Original entry on oeis.org

7, 92, 186, 423, 994, 1343, 2369, 3683, 5134, 6012, 7831, 8955, 11596, 12428, 15517, 16802, 21148, 28720, 31929, 33321, 41807, 44778, 51856, 51253, 57466, 57845, 82063, 88015, 95281, 97050, 117916, 127225, 130025, 135180, 165423, 161927, 176915, 183609, 193132, 202180, 228212, 228056, 236849

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..2500
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[];
for i1 from 1 to 300 do
    p:=ithprime(i1);
    if (p mod 8) = 7 then
        ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
        for j from 1 to p-1 do
            if legendre(j, p)=1 then
                q:=q+j;
                if jA282039, A282040, A282041, A282039 again, A282042, A282043

Table[Table[Mod[a^2, p], {a, 1, (p-1)/2}]//Total, {p, Select[Prime[Range[100]], Mod[#, 8] == 7 &]}] (* Vincenzo Librandi, Feb 21 2017 *)

A282042 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic nonresidues mod p that are > p/2.

Original entry on oeis.org

11, 128, 219, 520, 1176, 1348, 2221, 3310, 4766, 6106, 8034, 8271, 10049, 12443, 14613, 15193, 21012, 27486, 26814, 30664, 39248, 39318, 41699, 48888, 46052, 52595, 74613, 72878, 78599, 85768, 107895, 103643, 111125, 111195, 130497, 145619, 148490, 160159, 169503, 166856, 180406, 194204

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..1000
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 7 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282039, A282040, A282041, A282039 again, A282042, A282043

sum[p_]:= Total[If[#>p/2 && JacobiSymbol[#, p] != 1, #, 0]& /@ Range[p-1]];
sum /@ Select[Range[7, 1000, 8], PrimeQ] (* Jean-François Alcover, Aug 31 2018 *)

A282722 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p that are > p/2.

Original entry on oeis.org

0, 9, 44, 293, 461, 758, 1022, 1799, 2530, 3171, 4778, 5068, 7662, 8470, 9993, 14097, 16674, 19467, 20755, 25701, 29042, 34471, 37506, 40661, 45066, 48541, 54324, 54224, 58135, 60351, 68593, 75432, 74014, 83881, 85900, 98518, 112000, 117443, 122241, 132125, 143322, 151299, 163180, 161975, 181191

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..1000
Christian Aebi, Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282721 - A282727
# 2nd program
A282722 := proc(n)
    local p,ar;
    p := A007520(n) ;
    a := 0 ;
    for r from (p+1)/2 to p do
        if numtheory[legendre](r,p) = 1 then
            a := a+r ;
        end if;
    end do:
    a ;
end proc:
seq(A282722(n),n=1..10) ; # R. J. Mathar, Apr 07 2017

b[1] = 3; b[n_] := b[n] = Module[{p}, p = NextPrime[b[n - 1]]; While[Mod[p, 8] != 3, p = NextPrime[p]]; p];
a[n_] := Module[{p, q, r}, p = b[n]; q = 0; For[r = (p + 1)/2, r <= p, r++, If[KroneckerSymbol[r, p] == 1, q = q + r]]; q];
Array[a, 45] (* Jean-François Alcover, Nov 27 2017, after R. J. Mathar *)

A282723 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p.

Original entry on oeis.org

1, 22, 76, 430, 767, 1072, 1577, 2675, 3930, 4587, 6520, 7518, 10761, 12258, 14809, 19527, 23025, 26811, 29148, 35247, 41900, 47844, 52771, 57938, 61377, 66944, 73845, 76568, 79940, 83941, 94088, 102237, 104781, 114744, 117470, 134498, 152678, 161389, 167881, 181249, 193377, 204075, 221598, 228185

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..2500
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282721 - A282727

Table[Table[Mod[a^2, p], {a, 1, (p-1)/2}]//Total, {p, Select[Prime[Range[100]], Mod[#, 8] == 3 &]}] (* Vincenzo Librandi, Feb 21 2017 *)

A282724 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic nonresidues mod p that are < p/2.

Original entry on oeis.org

0, 2, 13, 94, 129, 247, 306, 555, 745, 999, 1579, 1555, 2466, 2653, 3059, 4581, 5430, 6351, 6658, 8409, 9087, 11158, 11996, 12858, 14814, 15788, 17880, 17277, 18950, 19481, 22400, 24876, 23518, 27448, 28115, 32285, 36743, 38269, 39851, 43111, 47406, 50055, 53683, 51645, 58274, 66410, 65119, 76013, 80465

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..1000
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282721 - A282727
# 2nd program
A282724 := proc(n)
    local p,a,r;
    p := A007520(n) ;
    a := 0 ;
    for r from 1 to (p-1)/2 do
        if numtheory[legendre](r,p) <> 1 then
            a := a+r ;
        end if;
    end do:
    a ;
end proc:
seq(A282724(n),n=1..10) ; # R. J. Mathar, Apr 07 2017

b[1] = 3; b[n_] := b[n] = Module[{p}, p = NextPrime[b[n - 1]]; While[Mod[p, 8] != 3, p = NextPrime[p]]; p];
a[n_] := Module[{p, q, r}, p = b[n]; q = 0; For[r = 1, r <= (p - 1)/2, r++, If[KroneckerSymbol[r, p] != 1, q = q + r]]; q];
Array[a, 50] (* Jean-François Alcover, Nov 27 2017, after R. J. Mathar *)

A282726 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic nonresidues mod p .

Original entry on oeis.org

2, 33, 95, 473, 944, 1139, 1826, 2996, 4585, 5004, 6683, 8413, 11394, 13393, 16566, 20376, 23946, 27804, 30883, 36384, 45671, 50059, 56040, 62357, 62874, 69559, 75486, 81635, 82795, 88050, 97183, 104166, 112030, 118159, 120925, 138193, 156613, 167066, 173670, 187262, 196026, 206796, 226333, 242750

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..2500
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282721 - A282727

Table[Table[Mod[a^2, p/2], {a, 1, (p-1)}]//Total, {p, Select[Prime[ Range[ 200]], Mod[#, 8] == 3 &]}] (* Vincenzo Librandi, Feb 22 2017 *)

cp's OEIS Frontend

A282043 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic nonresidues mod p.

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A282727 Let p = n-th prime == 3 mod 8; a(n) = (sum of quadratic residues mod p that are < p/2) + (sum of all quadratic residues mod p).

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A282039 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p that are < p/2.

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Maple

Mathematica

A282040 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p that are > p/2.

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Maple

Mathematica

A282041 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p.

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Author

Keywords

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Programs

Maple

Mathematica

A282042 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic nonresidues mod p that are > p/2.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A282722 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p that are > p/2.

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Maple

Mathematica

A282723 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p.

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Author

Keywords

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Programs

Maple

Mathematica

A282724 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic nonresidues mod p that are < p/2.

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