A282721 -id:A282721

A282043

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..42.

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

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; # A282039, A282040, A282041, A282039 again, A282042, A282043

MATHEMATICA

sqnr[p_] := Select[Range[p-1], JacobiSymbol[#, p] != 1&] // Total;

sqnr /@ Select[Prime[Range[200]], Mod[#, 8] == 7&] (* Jean-François Alcover, Aug 30 2018 *)

CROSSREFS

Cf. A282035-A282042 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

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).

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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).

LINKS

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.

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

Th:=[op(Th), q+ql];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; Th; # A282721 - 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

MATHEMATICA

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 *)

PROG

(Python)

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

CROSSREFS

Cf. A282035-A282043 and A282721-A282726.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

A282039

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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).

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; # A282039, 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

MATHEMATICA

sum[p_]:= Total[If[#<p/2 && JacobiSymbol[#, p] != 1, #, 0]& /@ Range[p-1]];

sum /@ Select[Range[7, 1100, 8], PrimeQ] (* Jean-François Alcover, Aug 31 2018 *)

CROSSREFS

Cf. A282035-A282043 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

A282040

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..43.

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

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; # A282039, A282040, A282041, A282039 again, A282042, A282043

MATHEMATICA

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 *)

CROSSREFS

Cf. A282035-A282043 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

A282041

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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).

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; # A282039, A282040, A282041, A282039 again, A282042, A282043

MATHEMATICA

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 *)

CROSSREFS

Cf. A282035-A282043 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

A282042

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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).

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; # A282039, A282040, A282041, A282039 again, A282042, A282043

MATHEMATICA

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 *)

CROSSREFS

Cf. A282035-A282043 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

A282722

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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.

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

Th:=[op(Th), q+ql];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; Th; # A282721 - 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

MATHEMATICA

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 *)

CROSSREFS

Cf. A282035-A282043 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

A282723

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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).

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

Th:=[op(Th), q+ql];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; Th; # A282721 - A282727

MATHEMATICA

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 *)

CROSSREFS

Cf. A282035-A282043 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

A282724

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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).

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

Th:=[op(Th), q+ql];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; Th; # A282721 - 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

MATHEMATICA

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 *)

CROSSREFS

Cf. A282035-A282043 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

A282726

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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).

MAPLE

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 j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

else

n:=n+j;

if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

fi;

od;

Ql:=[op(Ql), ql];

Qu:=[op(Qu), qu];

Q:=[op(Q), q];

Nl:=[op(Nl), nl];

Nu:=[op(Nu), nu];

N:=[op(N), n];

Th:=[op(Th), q+ql];

fi;

od:

Ql; Qu; Q; Nl; Nu; N; Th; # A282721 - A282727

MATHEMATICA

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 *)

CROSSREFS

Cf. A282035-A282043 and A282721-A282727.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved