cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Aug 02 2004

Keywords

Comments

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

Examples

			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
		

Crossrefs

See A226520 for another version.
Cf. A068717.

Programs

  • Haskell
    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
  • Maple
    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
  • Mathematica
    Flatten[ Table[ JacobiSymbol[ Range[ Prime[n]], Prime[n]], {n, 2, 8}]]

Formula

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