A096638 -id:A096638 - cp's OEIS frontend

A096636 Smallest prime p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

5, 7, 19, 79, 331, 751, 1171, 7459, 10651, 18379, 90931, 78439, 399499, 644869, 2631511, 1427911, 4355311, 5715319, 49196359, 43030381, 163384621, 249623581, 452980999, 1272463669, 505313251

Offset: 0

Robert G. Wilson v, Jun 24 2004

nonn

Same as smallest prime p with property that the Legendre symbol (p|q) = 1 for the first n odd primes q = prime(k+1), k = 1, 2, ..., n, and (p|q) = -1 for q = prime(n+2). - T. D. Noe, Mar 06 2013

			Let f(p) = list of Legendre(p|q) for q = 3,5,7,11,13,...
Then f(3), f(5), f(7), f(11), ... are:
p=3: 0, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, 1, -1, ...
p=5: -1, 0, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, ...
p=7: 1, -1, 0, -1, -1, -1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, ...
p=11: -1, 1, 1, 0, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, -1, 1, 1, 1, ...
p=13: 1, -1, -1, -1, 0, 1, -1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, -1, 1, -1, -1, ...
p=17: -1, -1, -1, -1, 1, 0, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, 1, 1, ...
p=19: 1, 1, -1, -1, -1, 1, 0, -1, -1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, ...
p=5 is the first list that begins with -1, so a(0) = 5,
p=7 is the first list that begins 1, -1, so a(1) = 7,
p=19 is the first list that begins 1, 1, -1, so a(2) = 19.

Cf. A094929, A222756 (p and q switched).

See also A096637, A096638, A096639, A096640. - Jonathan Sondow, Mar 07 2013

f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; t = Table[0, {50}]; Do[p = Prime[n]; a = f[p]; If[ t[[ PrimePi[a]]] == 0, t[[ PrimePi[a]]] = p; Print[ PrimePi[a], " = ", p]], {n, 10^9}]

Better definition from T. D. Noe, Mar 06 2013
Entry revised by N. J. A. Sloane, Mar 06 2013
Simpler definition from Jonathan Sondow, Mar 06 2013

A096634 Let p = n-th prime == 5 (mod 8) (A007521); a(n) = smallest prime q such that p is not a square mod q.

Original entry on oeis.org

3, 5, 3, 5, 3, 7, 3, 11, 3, 5, 3, 7, 3, 7, 3, 5, 3, 3, 7, 5, 3, 5, 13, 3, 3, 11, 3, 5, 3, 7, 3, 3, 13, 5, 5, 3, 3, 3, 7, 5, 5, 3, 5, 3, 7, 3, 7, 5, 3, 5, 3, 5, 3, 5, 3, 3, 3, 11, 11, 5, 3, 13, 5, 3, 17, 3, 7, 5, 3, 3, 7, 11, 7, 3, 3, 5, 3, 3, 3, 7, 5, 3, 3, 3, 11, 3, 13, 5, 3, 3, 7, 3, 3, 11, 5, 3, 3, 5, 3

Offset: 1

Robert G. Wilson v, Jun 24 2004

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A094928, A096633, A096635, A096638.

g:= proc(n) local p;
  p:= 1;
  do
    p:= nextprime(p);
    if numtheory:-quadres(n,p) = -1 then return p fi
  od
end proc:
map(g, select(isprime, [seq(i,i=5..10000,8)])); # Robert Israel, Apr 17 2023

f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; f /@ Select[ Prime[ Range[435]], Mod[ #, 8] == 5 &]

cp's OEIS Frontend

A096636 Smallest prime p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

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