A222756 -id:A222756 - 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

A237436 Least prime p > prime(n+1) such that p is a square mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1).

Original entry on oeis.org

7, 19, 79, 331, 751, 1171, 7459, 10651, 18379, 78439, 78439, 399499, 644869, 1427911, 1427911, 4355311, 5715319, 43030381, 43030381, 163384621

Offset: 1

Jonathan Sondow, Feb 15 2014

nonn

Least prime p > prime(n+1) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1).

Least odd prime p such that the Legendre symbol (p|q) = 1 for q = 3, 5, 7, 11, ..., prime(n+1).

			Let f(p) = list of Legendre (p|q) for q = 3, 5, 7, 11, 13, 17, 19, 23, ...
Then f(p) is
p=3: 0, -1, -1, 1, 1, -1, -1, 1, ...
p=5: -1, 0, -1, 1, -1, -1, 1, -1, ...
p=7: 1, -1, 0, -1, -1, -1, 1, -1, ...
p=11: -1, 1, 1, 0, -1, -1, 1, -1, ...
p=13: 1, -1, -1, -1, 0, 1, -1, 1, ...
p=17: -1, -1, -1, -1, 1, 0, 1, -1, ...
p=19: 1, 1, -1, -1, -1, 1, 0, -1, ...
f(7) is the first list that begins with 1, so a(1) = 7.
f(19) is the first list that begins with 1, 1, so a(2) = 19.

Wikipedia, Legendre symbol
Wikipedia, Quadratic residue

Cf. A222756 (p and q switched), A237437.

Table[p = Prime[n+2]; While[Length[Select[Prime[Range[2, n + 1]], JacobiSymbol[p, #] == 1 &]] < n, p = NextPrime[p]]; p, {n, 1, 18}]

a(n) = a(n+1) if and only if Legendre (a(n)|prime(n+2)) = 1.
a(n) <= A096636(n).
a(n) < A096636(n) if and only if a(n) = a(n+1).
a(n) = A096636(n) if and only if Legendre (a(n)|prime(n+2)) = -1.

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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