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A237437 Least prime p > prime(n+1) such that p is not a square mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1).

%I A237437 #8 Mar 16 2023 12:22:57
%S A237437 5,17,17,17,83,167,167,227,2273,5297,5297,69467,69467,116387,348563,
%T A237437 348563,2004917,5472953,8062073,8062073
%N A237437 Least prime p > prime(n+1) such that p is not a square mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1).
%C A237437 Least prime p > prime(n+1) such that p is a quadratic nonresidue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1).
%C A237437 Least odd prime p such that the Legendre symbol (p|q) = -1 for q = 3, 5, 7, 11, ..., prime(n+1).
%H A237437 Wipawee Tangjai, Kodchaphon Wanichang, Montathip Srikao, and Punyanuch Kheawkrai, <a href="https://doi.org/10.28924/2291-8639-21-2023-24">A Congruent Property of Gibonacci Number Modulo Prime</a>, Int'l. J. Analysis Appl. (2023), Vol. 21, No. 24.
%H A237437 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_symbol">Legendre symbol</a>
%H A237437 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quadratic_residue">Quadratic residue</a>
%F A237437 a(n) = a(n+1) if and only if Legendre (a(n)|prime(n+2)) = -1.
%e A237437 Let f(p) = list of Legendre (p|q) for q = 3, 5, 7, 11, 13, 17, 19, 23, ...
%e A237437 Then f(p) is
%e A237437 p=3: 0, -1, -1, 1, 1, -1, -1, 1, ...
%e A237437 p=5: -1, 0, -1, 1, -1, -1, 1, -1, ...
%e A237437 p=7: 1, -1, 0, -1, -1, -1, 1, -1, ...
%e A237437 p=11: -1, 1, 1, 0, -1, -1, 1, -1, ...
%e A237437 p=13: 1, -1, -1, -1, 0, 1, -1, 1, ...
%e A237437 p=17: -1, -1, -1, -1, 1, 0, 1, -1, ...
%e A237437 p=19: 1, 1, -1, -1, -1, 1, 0, -1, ...
%e A237437 f(5) is the first list that begins with -1, so a(1) = 5.
%e A237437 f(17) is the first list that begins with -1, -1, so a(2) = 17.
%t A237437 Table[p = Prime[n + 2]; While[Length[Select[Prime[Range[2, n + 1]], JacobiSymbol[p, #] == -1 &]] < n, p = NextPrime[p]]; p, {n, 1, 20}]
%Y A237437 Cf. A237436.
%K A237437 nonn
%O A237437 1,1
%A A237437 _Jonathan Sondow_, Feb 15 2014