A063987 -id:A063987 - cp's OEIS frontend

A096008 Irregular triangle read by rows where n-th row contains all quadratic residues (including zero) mod n.

0, 0, 1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 3, 4, 0, 1, 2, 4, 0, 1, 4, 0, 1, 4, 7, 0, 1, 4, 5, 6, 9, 0, 1, 3, 4, 5, 9, 0, 1, 4, 9, 0, 1, 3, 4, 9, 10, 12, 0, 1, 2, 4, 7, 8, 9, 11, 0, 1, 4, 6, 9, 10, 0, 1, 4, 9, 0, 1, 2, 4, 8, 9, 13, 15, 16, 0, 1, 4, 7, 9, 10, 13, 16, 0, 1, 4, 5, 6, 7, 9, 11, 16, 17, 0, 1, 4, 5, 9, 16

Offset: 1

Cino Hilliard, Jul 20 2004

			The table starts:
  [1]  [0]
  [2]  [0, 1]
  [3]  [0, 1]
  [4]  [0, 1]
  [5]  [0, 1, 4]
  [6]  [0, 1, 3, 4]
  [7]  [0, 1, 2, 4]
  [8]  [0, 1, 4]
  [9]  [0, 1, 4, 7]
  [10] [0, 1, 4, 5, 6, 9]
  ...

T. D. Noe, Rows n = 1..100, flattened
Eric Weisstein's World of Mathematics, Quadratic Residue.

Cf. A046071 (without zeros), A000224 (row lengths), A063987.
Last elements of rows give A047210.
Row sums give A165909.

a096008 n k = a096008_tabf !! (n-1) !! (k-1)
a096008_row n = a096008_tabf !! (n-1)
a096008_tabf = [0] : map (0 :) a046071_tabf
-- Reinhard Zumkeller, May 10 2015

q := n -> sort(convert({seq(i^2 mod n, i=0..n-1)}, list));
# N. J. A. Sloane, Feb 09 2011
# Alternative:
QR := (a, n) -> NumberTheory:-QuadraticResidue(a, n):
for n from 1 to 10 do print(select(a -> 1 = QR(a, n), [seq(0..n-1)])) od:
# Peter Luschny, Jun 02 2024

row[n_] := Table[PowerMod[k, 2, n], {k, 0, n-1}] // Union; Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Sep 09 2013 *)
ResourceFunction["QuadraticResidues"] /@ Range[20] // Flatten  (* Peter Luschny, May 23 2024 *)

T(n) = {local(v,r,i,j,k); v=vector(n,i,0); for(i=0,floor(n/2),v[i^2%n+1]=1); k=sum(i=1,n,v[i]); j=0; r=vector(k); for(i=1,n, if(v[i], j++; r[j]=i-1)); r}

for n in range(1, 11): print(quadratic_residues(n)) # Peter Luschny, Jun 02 2024

Edited by Franklin T. Adams-Watters, Nov 07 2006

A130290 Number of nonzero quadratic residues modulo the n-th prime.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158

Offset: 1

M. F. Hasler, May 21 2007

Row lengths for formatting A063987 as a table: The number of nonzero quadratic residues modulo a prime p equals floor(p/2), or (p-1)/2 if p is odd. The number of squares including 0 is (p+1)/2, if p is odd (rows prime(i) of A096008 formatted as a table). In fields of characteristic 2, all elements are squares. For any m > 0, floor(m/2) is the number of even positive integers less than or equal to m, so a(n) also equals the number of even positive integers less than or equal to the n-th prime. For all n > 0, A130290(n+1) = A005097(n) = A102781(n+1) = A102781(n+1) = A130291(n+1)-1 = A111333(n+1)-1 = A006254(n)-1.
From Vladimir Shevelev, Jun 18 2016: (Start)
a(1)+2 and, for n >= 2, a(n)+1 is the smallest k such that there exists 0 < k_1 < k with the condition k_1^2 == k^2 (mod prime(n)).
Indeed, for n >= 2, if prime(n) = 4*t+1 then k = 2*t+1 = a(n)+1, since (2*t+1)^2 == (2*t)^2 (mod prime(n)) and there cannot be a smaller value of k; if prime(n) = 4*t-1, then k = 2*t = a(n)+1, since (2*t)^2 == (2*t-1)^2 (mod prime(n)). (End)
a(n) is the number of pairs (a,b) such that a + b = prime(n) with 1 <= a <= b. - Nicholas Leonard, Oct 02 2022

			a(1)=1 since the only nonzero element of Z/2Z equals its square.
a(3)=2 since 1=1^2=(-1)^2 and 4=2^2=(-2)^2 are the only nonzero squares in Z/5Z.
a(1000000) = 7742931 = (prime(1000000)-1)/2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Quadratic Residue
Wikipedia, Quadratic Residue

Essentially the same as A005097.
Cf. A102781 (Number of even numbers less than the n-th prime), A063987 (quadratic residues modulo the n-th prime), A006254 (Numbers n such that 2n-1 is prime), A111333 (Number of odd numbers <= n-th prime), A000040 (prime numbers), A130291.
Appears in A217983. - Johannes W. Meijer, Oct 25 2012

[Floor((NthPrime(n))/2): n in [1..60]]; // Vincenzo Librandi, Jan 16 2013

A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: A000040 := proc(n): ithprime(n) end: seq(A130290(n), n=1..55); # Johannes W. Meijer, Oct 25 2012

Quotient[Prime[Range[66]], 2] (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)

PARI
```
A130290(n) = prime(n)>>1
```

from sympy import prime
def A130290(n): return prime(n)//2  # Chai Wah Wu, Jun 04 2022

a(n) = floor( A000040(n)/2 ) = #{ even positive integers <= A000040(n) }
a(n) = A055034(A000040(n)), n>=1. - Wolfdieter Lang, Sep 20 2012
a(n) = A005097(n-[n>1]) = A005097(max(n-1,1)). - M. F. Hasler, Dec 13 2019

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

Original entry on oeis.org

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

A048152 Triangular array T read by rows: T(n,k) = k^2 mod n, for 1 <= k <= n, n >= 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 4, 3, 4, 1, 0, 1, 4, 2, 2, 4, 1, 0, 1, 4, 1, 0, 1, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1, 0

Offset: 1

Clark Kimberling

			Rows:
  0;
  1, 0;
  1, 1, 0;
  1, 0, 1, 0;
  1, 4, 4, 1, 0;
  1, 4, 3, 4, 1, 0;

T. D. Noe, Rows n = 1..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Quadratic Residue

Cf. A060036.
Cf. A225126 (central terms).
Cf. A070430 (row 5), A070431 (row 6), A053879 (row 7), A070432 (row 8), A008959 (row 10), A070435 (row 12), A070438 (row 15), A070422 (row 20).
Cf. A046071 (in ascending order, without zeros and duplicates).
Cf. A063987 (for primes, in ascending order, without zeros and duplicates).

a048152 n k = a048152_tabl !! (n-1) !! (k-1)
a048152_row n = a048152_tabl !! (n-1)
a048152_tabl = zipWith (map . flip mod) [1..] a133819_tabl
-- Reinhard Zumkeller, Apr 29 2013

Flatten[Table[PowerMod[k,2,n],{n,15},{k,n}]] (* Harvey P. Dale, Jun 20 2011 *)

T(n,k) = A133819(n,k) mod n, k = 1..n. - Reinhard Zumkeller, Apr 29 2013

T(n,k) = (T(n,k-1) + (2k+1)) mod n. - Andrés Ventas, Apr 06 2021

A046071 Triangle of nonzero quadratic residues mod n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 3, 4, 1, 2, 4, 1, 4, 1, 4, 7, 1, 4, 5, 6, 9, 1, 3, 4, 5, 9, 1, 4, 9, 1, 3, 4, 9, 10, 12, 1, 2, 4, 7, 8, 9, 11, 1, 4, 6, 9, 10, 1, 4, 9, 1, 2, 4, 8, 9, 13, 15, 16, 1, 4, 7, 9, 10, 13, 16, 1, 4, 5, 6, 7, 9, 11, 16, 17, 1, 4, 5, 9, 16, 1, 4, 7, 9, 15, 16, 18, 1, 3, 4, 5, 9, 11, 12

Offset: 2

Eric W. Weisstein

Rows start with 1's.

			1,
1,
1,
1, 4,
1, 3, 4,
1, 2, 4,
1, 4,
1, 4, 7,
1, 4, 5, 6, 9,
1, 3, 4, 5, 9,
1, 4, 9,
1, 3, 4, 9, 10, 12,
1, 2, 4, 7, 8, 9, 11
1, 4, 6, 9, 10,
- _Geoffrey Critzer_, Apr 03 2015

T. D. Noe, Rows n = 1..100, flattened
Eric Weisstein's World of Mathematics, Quadratic Residue.

Cf. A105612 (row lengths), A165909 (row sums), A372651 (row products).

Cf. A096008 (including zeros), A063987.

import Data.List (sort, nub, genericIndex)
a046071 n k = genericIndex a046071_tabf (n-2) !! (k-1)
a046071_row n = genericIndex a046071_tabf (n-2)
a046071_tabf = f [1] 2 3 where
   f qs@(q:_) i j = ys : f ((q + j) : qs) (i + 1) (j + 2) where
                    ys = nub $ sort $ filter (> 0) $ map (flip mod i) qs
-- Reinhard Zumkeller, May 10 2015

seq(op(select(numtheory:-quadres=1,[$1..n-1],n)),n=2..30); # Robert Israel, Apr 03 2015

residueQ[n_, k_] := Length[ Select[ Range[ Floor[k/2]]^2, Mod[#, k] == n & , 1]] == 1; row[n_] := Select[ Range[n-1], residueQ[#, n]& ]; Table[row[n], {n, 2, 22}] // Flatten (* Jean-François Alcover, Oct 23 2012 *)
row[n_] := Table[PowerMod[k, 2, n], {k, 0, n-1}] // Union // Rest; Table[row[n], {n, 2, 22}] // Flatten (* Jean-François Alcover, Jul 07 2019 *)

residue(n,m)={local(r);r=0;for(i=0,floor(m/2),if(i^2%m==n,r=1));r} \\ Michael B. Porter, May 03 2010

for n in range(2, 16): print(quadratic_residues(n)[1:]) # Peter Luschny, Jun 02 2024

Edited by Franklin T. Adams-Watters, Nov 07 2006

A130291 Number of quadratic residues (including 0) modulo the n-th prime.

Original entry on oeis.org

2, 2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157

Offset: 1

M. F. Hasler, May 21 2007

The number of squares (quadratic residues including 0) modulo a prime p (sequence A096008 with every "1" prefixed by a "0") equals 1+floor(p/2), or ceiling(p/2) = (p+1)/2 if p is odd. (In fields of characteristic 2, all elements are squares.) See A130290(n)=A130291(n)-1 for number of nonzero residues. For all n>0, A130291(n+1) = A111333(n+1) = A006254(n) = A005097(n)-1 = A102781(n+1)-1 = A102781(n+1)-1 = A130290(n+1)-1.

			a(1)=2 since both elements of Z/2Z are squares.
a(3)=0 since 0=0^2, 1=1^2=(-1)^2 and 4=2^2=(-2)^2 are squares in Z/5Z.
a(1000000) = 7742932 = (p[1000000]+1)/2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Quadratic Residue
Wikipedia, Quadratic Residue

Essentially the same as A006254.

Cf. A005097 (Odd primes - 1)/2, A102781 (Integer part of n#/(n-2)#/2#), A102781 (Number of even numbers less than the n-th prime), A063987 (quadratic residues modulo the n-th prime), A006254 (Numbers n such that 2n-1 is prime), A111333 (Number of odd numbers <= n-th prime), A000040 (prime numbers), A130290 (number of nonzero residues modulo primes).

[Floor((NthPrime(n))/2)+1: n in [1..60]]; // Vincenzo Librandi, Jan 16 2013

Quotient[Prime[Range[200]], 2] + 1 (* Vincenzo Librandi, Jan 16 2013 *)

PARI
```
A130291(n) = 1+prime(n)>>1
```

a(n) = floor( A000040(n)/2 )+1

A230077 Table a(n,m) giving in row n all k from {1, 2, ..., prime(n)-1} for which the Legendre symbol (k/prime(n)) = +1, for odd prime(n).

Original entry on oeis.org

1, 1, 4, 1, 4, 2, 1, 4, 9, 5, 3, 1, 4, 9, 3, 12, 10, 1, 4, 9, 16, 8, 2, 15, 13, 1, 4, 9, 16, 6, 17, 11, 7, 5, 1, 4, 9, 16, 2, 13, 3, 18, 12, 8, 6, 1, 4, 9, 16, 25, 7, 20, 6, 23, 13, 5, 28, 24, 22, 1, 4, 9, 16, 25, 5, 18, 2, 19, 7, 28, 20, 14, 10, 8

Offset: 2

Wolfdieter Lang, Oct 25 2013

The length of row n is r(n) = (prime(n) - 1)/2, with prime(n) = A000040(n), n >= 2.
If k from {1, 2, ..., p-1} appears in row n then the Legendre symbol (k/prime(n)) = +1 otherwise it is -1.
The Legendre symbol (k/p), p an odd prime and gcd(k,p) = 1, is +1 if there exists an integer x with x^2 == k (mod p) and -1 otherwise. It is sufficient to consider k from {1, 2, ..., p-1} (gcd(0,p) = p, not 1) because (k/p) = ((k + l*p)/p) for integer l. Because (p - x)^2 == x^2 (mod p), it is also sufficient to consider only x^2 from {1^2, 2^2, ..., ((p-1)/2)^2} which are pairwise incongruent modulo p. See the Hardy-Wright reference. p. 68-69.
For odd primes p one has for the Legendre symbol ((p-1)/p) =  (-1/p) = (-1)^(r(n)) (see above for the row length r(n), and theorem 82, p. 69 of Hardy-Wright), and this is +1 for prime p == 1 (mod 4) and -1 for p == 3 (mod 4). Therefore k = p-1 appears in row n iff p = prime(n) is from A002144 = 5, 13, 17, 29, 37, 41,...
For n>=4 (prime(n)>=7) at least one of the integers 2, 3, or 6 appears in every row. - Geoffrey Critzer, May 01 2015

			The irregular table a(n,m) begins (here p(n)=prime(n)):
n, p(n)\m 1 2 3  4  5  6   7   8   9  10  11  12  13  14  15
2,   3:   1
3,   5:   1 4
4,   7:   1 4 2
5,  11:   1 4 9  5  3
6,  13:   1 4 9  3 12 10
7,  17:   1 4 9 16  8  2  15  13
8,  19:   1 4 9 16  6 17  11   7   5
9,  23:   1 4 9 16  2 13   3  18  12   8   6
10, 29:   1 4 9 16 25  7  20   6  23  13   5  28  24  22
11, 31    1 4 9 16 25  5  18   2  19   7  28  20  14  10   8
...
Row n=12, p(n)=37: 1, 4, 9, 16, 25, 36, 12, 27, 7, 26, 10, 33, 21, 11, 3, 34, 30, 28.
Row n=13, p(n)=41: 1, 4, 9, 16, 25, 36, 8, 23, 40, 18, 39, 21, 5, 32, 20, 10, 2, 37, 33, 31.
(2/p) = +1 for n=4, p(4) = 7; p(7) = 17, p(9) = 23, p(11) = 31, p(13) = 41, ... This leads to A001132 (primes 1 or 7 (mod 8)).
4 = 5 - 1 appears in row n=3 for p(3)=5 because 5 is from A002144. 10 cannot appear in row 5 for p(5)=11 because 11 == 3 (mod 4), hence 11 is not in A002144.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, 2003.

Alois P. Heinz, Rows n = 2..100, flattened

Cf. A000040, A002144, A063987.

T:= n-> (p-> seq(irem(m^2, p), m=1..(p-1)/2))(ithprime(n)):
seq(T(n), n=2..12);  # Alois P. Heinz, May 07 2015

Table[Table[Mod[a^2, p], {a, 1, (p - 1)/2}], {p,
Prime[Range[2, 20]]}] // Grid (* Geoffrey Critzer, Apr 30 2015 *)

a(n,m) = m^2 (mod prime(n)), n >= 2, where prime(n) = A000040(n), m = 1, 2, ..., (prime(n) - 1)/2.

A269595 Irregular triangle in which n-th row the gives quadratic residues prime(n)- m modulo prime(n), for m from {1, 2, ..., prime(n)-1}, in increasing order.

Original entry on oeis.org

1, 2, 1, 4, 3, 5, 6, 2, 6, 7, 8, 10, 1, 3, 4, 9, 10, 12, 1, 2, 4, 8, 9, 13, 15, 16, 2, 3, 8, 10, 12, 13, 14, 15, 18, 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22, 1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28

Offset: 1

Wolfdieter Lang, Mar 06 2016

The length of row 1 is 1 and of row n, n >= 2, is (prime(n)-1)/2, where prime(n) = A000040(n).

			The irregular triangle T(n, k) begins (P(n) is here prime(n)):
n, P(n)\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14
1,   2:   1
2,   3:   2
3,   5:   1  4
4,   7:   1  2  4
5,  11:   1  3  4  5  9
6:  13:   1  3  4  9 10 12
7,  17:   1  2  4  8  9 13 15 16
8,  19:   1  4  5  6  7  9 11 16 17
9,  23:   1  2  3  4  6  8  9 12 13 16 18
10, 29:   1  4  5  6  7  9 13 16 20 22 23 24 25 28
...

Cf. A000040, A063987.

t = Table[Select[Range[Prime@ n - 1], JacobiSymbol[#, Prime@ n] == 1 &], {n, 10}]; Table[Prime@ n - t[[n, (Prime@ n - 1)/2 - k + 1]], {n, Length@ t}, {k, (Prime@ n - 1)/2}] /. {} -> 1 // Flatten (* Michael De Vlieger, Mar 31 2016, after Jean-François Alcover at A063987 *)

For n = 1, prime(1) = 2: 1, and for odd primes n >= 2: the increasing values of m from {1, 2, ..., p-1} with the Legendre symbol (-m/prime(n)) = + 1.

T(n, k) = prime(n) - A063987(n,(prime(n)-1)/2-k+1). k=1..(prime(n)-1)/2, for n >= 2, and T(1, 1) = 1.

A278580 Numbers n such that Jacobi(n,23) = 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 25, 26, 27, 29, 31, 32, 35, 36, 39, 41, 47, 48, 49, 50, 52, 54, 55, 58, 59, 62, 64, 70, 71, 72, 73, 75, 77, 78, 81, 82, 85, 87, 93, 94, 95, 96, 98, 100, 101, 104, 105, 108, 110, 116, 117, 118, 119, 121, 123, 124, 127, 128, 131, 133, 139, 140, 141, 142, 144, 146

Offset: 1

N. J. A. Sloane, Nov 29 2016

Important for the study of Ramanujan numbers A000594.

The first 11 terms, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, are the quadratic residues mod 23 (see row 23 of A063987).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A010385, A000594, A063987, A278579.

LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,1,-1},{1,2,3,4,6,8,9,12,13,16,18,24},90] (* Harvey P. Dale, Jun 25 2020 *)

Vec(x*(1+x+x^2+x^3+2*x^4+2*x^5+x^6+3*x^7+x^8+3*x^9+2*x^10+5*x^11) / ((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)) + O(x^100)) \\ Colin Barker, Nov 30 2016

From Colin Barker, Nov 30 2016: (Start)
a(n+11) = a(n) + 23.
a(n) = a(n-1) + a(n-11) - a(n-12) for n>12.
G.f.: x*(1 +x +x^2 +x^3 +2*x^4 +2*x^5 +x^6 +3*x^7 +x^8 +3*x^9 +2*x^10 +5*x^11) / ((1 -x)^2*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 +x^8 +x^9 +x^10))
(End)

A010385 Squares mod 23.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18

Offset: 1

N. J. A. Sloane

Row 23 of A096008. Row 9 of A063987. A028736 (complement), A278580.

Join[{0}, Select[Range[22], JacobiSymbol[#, 23] == 1&]] (* Jean-François Alcover, Oct 07 2018 *)

[quadratic_residues(23)] # Zerinvary Lajos, May 24 2009

cp's OEIS Frontend

A096008 Irregular triangle read by rows where n-th row contains all quadratic residues (including zero) mod n.

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A130290 Number of nonzero quadratic residues modulo the n-th prime.

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

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A048152 Triangular array T read by rows: T(n,k) = k^2 mod n, for 1 <= k <= n, n >= 1.

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A046071 Triangle of nonzero quadratic residues mod n.

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A130291 Number of quadratic residues (including 0) modulo the n-th prime.

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