A237436 - OEIS

%I #4 Feb 15 2014 13:52:09

%S 7,19,79,331,751,1171,7459,10651,18379,78439,78439,399499,644869,

%T 1427911,1427911,4355311,5715319,43030381,43030381,163384621

%N 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).

%C 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).

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_symbol">Legendre symbol</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Quadratic_residue">Quadratic residue</a>

%F a(n) = a(n+1) if and only if Legendre (a(n)|prime(n+2)) = 1.

%F a(n) <= A096636(n).

%F a(n) < A096636(n) if and only if a(n) = a(n+1).

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

%e Let f(p) = list of Legendre (p|q) for q = 3, 5, 7, 11, 13, 17, 19, 23, ...

%e Then f(p) is

%e p=3: 0, -1, -1, 1, 1, -1, -1, 1, ...

%e p=5: -1, 0, -1, 1, -1, -1, 1, -1, ...

%e p=7: 1, -1, 0, -1, -1, -1, 1, -1, ...

%e p=11: -1, 1, 1, 0, -1, -1, 1, -1, ...

%e p=13: 1, -1, -1, -1, 0, 1, -1, 1, ...

%e p=17: -1, -1, -1, -1, 1, 0, 1, -1, ...

%e p=19: 1, 1, -1, -1, -1, 1, 0, -1, ...

%e f(7) is the first list that begins with 1, so a(1) = 7.

%e f(19) is the first list that begins with 1, 1, so a(2) = 19.

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

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

%K nonn

%O 1,1

%A _Jonathan Sondow_, Feb 15 2014