A097343 Triangle read by rows in which row n gives Legendre symbol (k,p) for 0<k<=p where p = n-th prime.

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

2,1

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

Links

Reinhard Zumkeller, Rows n = 2..75 of triangle, flattened

Haskell for Math, Number Theory Fundamentals

Wikipedia, Legendre symbol

Formula

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

Example

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

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}]]

Prog

(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

Crossrefs

See A226520 for another version.

Cf. A068717.

Sequence in context: A138019 A179850 A267919 * A188011 A040051 A108788

Adjacent sequences: A097340 A097341 A097342 * A097344 A097345 A097346

Keyword

sign,tabf

Author

Robert G. Wilson v, Aug 02 2004

Status

approved