A011582 -id:A011582 - cp's OEIS frontend

A097343 Triangle read by rows in which row n gives Legendre symbol (k,p) for 0

1, -1, 0, 1, -1, -1, 1, 0, 1, 1, -1, 1, -1, -1, 0, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 0, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 0, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, -1, 0, 1, 1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, -1, -1, 0, 1, -1, -1, 1, 1, 1, 1

Offset: 2

Robert G. Wilson v, Aug 02 2004

Row sums = 0. (p,k)==k^((p-1)/2) (mod p). For example, row n=4 of the triangle (for the 4th prime p = 7) reads: 1,1,-1,1,-1,-1,0 because 1^3==1, 2^3==1, 3^3==-1, 4^3==1, 5^3==-1, 6^3==-1, 7^3==0 (mod 7). - Geoffrey Critzer, Apr 18 2015

			1,-1,0 ; # A102283
1,-1,-1,1,0; # A080891
1,1,-1,1,-1,-1,0; # A175629
1,-1,1,1,1,-1,-1,-1,1,-1,0; # A011582

Reinhard Zumkeller, Rows n = 2..75 of triangle, flattened
Haskell for Math, Number Theory Fundamentals
Wikipedia, Legendre symbol

See A226520 for another version.

Cf. A068717.

a097343 n k = a097343_tabf !! (n-2) !! (k-1)
a097343_row n = a097343_tabf !! (n-2)
a097343_tabf =
   map (\p -> map (flip legendreSymbol p) [1..p]) $ tail a000040_list
legendreSymbol a p = if a' == 0 then 0 else twoSymbol * oddSymbol where
   a' = a `mod` p
   (s,q) = a' `splitWith` 2
   twoSymbol = if (p `mod` 8) `elem` [1,7] || even s then 1 else -1
   oddSymbol = if q == 1 then 1 else qrMultiplier * legendreSymbol p q
   qrMultiplier = if p `mod` 4 == 3 && q `mod` 4 == 3 then -1 else 1
   splitWith n p = spw 0 n where
      spw s t = if m > 0 then (s, t) else spw (s + 1) t'
                where (t', m) = divMod t p
-- See link.  Reinhard Zumkeller, Feb 02 2014

with(numtheory):
T:= n-> (p-> seq(jacobi(k, p), k=1..p))(ithprime(n)):
seq(T(n), n=2..15);  # Alois P. Heinz, Apr 19 2015

Flatten[ Table[ JacobiSymbol[ Range[ Prime[n]], Prime[n]], {n, 2, 8}]]

(p, p)=0, all others are +- 1.

A055088 Triangle of generalized Legendre symbols L(a/b) read by rows, with 1's for quadratic residues and 0's for quadratic non-residues.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Apr 18 2000

L(a/b) is 1 if an integer c exists such that c^2 is congruent to a (mod b) and 0 otherwise.

For every prime of the form 4k+1 (A002144) the row is symmetric and for every prime of the form 4k+3 (A002145) the row is "complementarily symmetric".

			The tenth row gives the quadratic residues and non-residues of 11 (see A011582) and the twelfth row gives the same information for 13 (A011583), with -1's replaced by zeros.
.
Triangle starts:
  [ 1] [1]
  [ 2] [1, 0]
  [ 3] [1, 0, 0]
  [ 4] [1, 0, 0, 1]
  [ 5] [1, 0, 1, 1, 0]
  [ 6] [1, 1, 0, 1, 0, 0]
  [ 7] [1, 0, 0, 1, 0, 0, 0]
  [ 8] [1, 0, 0, 1, 0, 0, 1, 0]
  [ 9] [1, 0, 0, 1, 1, 1, 0, 0, 1]
  [10] [1, 0, 1, 1, 1, 0, 0, 0, 1, 0]
  [11] [1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0]
  [12] [1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1]

Cf. A002144, A002145, A011582, A011583.

Each row interpreted as a binary number: A055094.

# See A054431 for one_or_zero and trinv.
with(numtheory,quadres); quadres_0_1_array := (n) -> one_or_zero(quadres((n-((trinv(n-1)*(trinv(n-1)-1))/2)), (trinv(n-1)+1)));

row[n_] := With[{rr = Table[Mod[k^2, n + 1], {k, 1, n}] // Union}, Boole[ MemberQ[rr, #]]& /@ Range[n]];
Array[row, 14] // Flatten (* Jean-François Alcover, Mar 05 2016 *)

def A055088_row(n) :
    Q = quadratic_residues(n+1)
    return [int(i in Q) for i in (1..n)]
for n in (1..14) : print(A055088_row(n))  # Peter Luschny, Aug 08 2012

A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).

Original entry on oeis.org

0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0

Offset: 0

Jianing Song, Dec 27 2018

Period 21: repeat [0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1].
Also a(n) = Kronecker symbol (21/n).
This sequence is one of the three non-principal real Dirichlet characters modulo 21. The other two are Jacobi or Kronecker symbols {(n/63)} (or {(-63/n)}) and {(n/147)} (or {(-147/n)}).

Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,0,-1,0,1,-1,0,1,-1).

Cf. A035203 (inverse Moebius transform).

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017(d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), this sequence (d=21), A322796 (d=24).

JacobiSymbol[Range[0, 100], 21] (* Paolo Xausa, Mar 19 2025 *)

PARI
```
a(n) = kronecker(n, 21)
```

a(n) = 1 for n == 1, 4, 5, 16, 17, 20 (mod 21); -1 for n == 2, 8, 10, 11, 13, 19 (mod 21); 0 for n that are not coprime with 21.
Completely multiplicative with a(p) = a(p mod 21) for primes p.
a(n) = A102283(n)*A175629(n).
a(n) = a(n+21) = -a(n) for all n in Z.
From Chai Wah Wu, Feb 18 2021: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-8) - a(n-9) + a(n-11) - a(n-12) for n > 11.
G.f.: -x*(x - 1)*(x + 1)*(x^8 - 2*x^7 + 2*x^6 + 2*x^2 - 2*x + 1)/(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1). (End)

A065103 A level 11 weight 5 form.

Original entry on oeis.org

1, 15, 82, 241, 626, 1230, 2400, 3855, 6643, 9390, 14641, 19762, 28560, 36000, 51332, 61681, 83520, 99645, 130320, 150866, 196800, 219615, 279842, 316110, 391251, 428400, 538084, 578400, 707280, 769980, 923522, 986895, 1200562

Offset: 1

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Nov 20 2001

Multiplicative because convolution of multiplicative functions. - Michael Somos, Jun 07 2005

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ken Ono, On the Circular Summation of the Eleventh Powers of Ramanujan's Theta Function, Journal of Number Theory, Volume 76, Issue 1, May 1999, Pages 62-65.

Cf. A011582.

a[n_]:= If[n < 0, 0, Sum[KroneckerSymbol[d, 11]*(n/d)^4, {d, Divisors[n]}]]; Table[a[n], {n,1,50}] (* G. C. Greubel, Apr 17 2018 *)

a(n) = sumdiv(n, d, kronecker(d, 11) * (n/d)^4); \\ Amiram Eldar, Jan 18 2025

a(n) = Sum_{d | n} Kronecker(d/11)*(n/d)^4.

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A097343 Triangle read by rows in which row n gives Legendre symbol (k,p) for 0

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A055088 Triangle of generalized Legendre symbols L(a/b) read by rows, with 1's for quadratic residues and 0's for quadratic non-residues.

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A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).

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A065103 A level 11 weight 5 form.

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