A094929 -id:A094929 - 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

A094928 Let p = n-th prime == 1 mod 8 (A007519); a(n) = smallest prime q such that p is not a square mod q.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 19 2004

			n=3, p = 73, a(3) = q = 5: Legendre(73,5) = -1.

M. Kneser, Quadratische Formen, Springer, 2002; see Hilfssatz 18.3.

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

Cf. A007519, A002224, A144294, A269704.

Subsequence of A094929.

f:= proc(p) local q;
     q:= 3:
     do
      if numtheory:-quadres(p,q) = -1 then return q fi;
      q:= nextprime(q);
     od;
end proc:
map(f, select(isprime, [seq(p,p=1..10000,8)])); # Robert Israel, May 06 2019

f[n_] := Prime[ Position[ JacobiSymbol[n, Select[Range[3, n - 1], PrimeQ[ # ] &]], -1][[1, 1]] + 1]; f /@ Select[ Prime[ Range[435]], Mod[ #, 8] == 1 &] (* Robert G. Wilson v, Jun 23 2004 *)

a(n) = A094929(A269704(n)). - Robert Israel, May 06 2019

More terms from Robert G. Wilson v, Jun 23 2004

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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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A094928 Let p = n-th prime == 1 mod 8 (A007519); a(n) = smallest prime q such that p is not a square mod q.

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