%I #18 Apr 10 2019 10:09:32

%S 0,7,7,23,17,47,31,79,0,17,71,167,97,223,127,41,23,359,199,439,241,31,

%T 41,89,337,727,0,839,449,137,73,1087,577,1223,647,1367,103,0,47,73,

%U 881,1847,967,0,151,2207,1151,2399,1249,113,193,401,0,3023,1567,191,0,71

%N Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists.

%C Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^2 = 2 iff n^2-2 has a prime factor > n; n is a solution mod p of x^2 = 2 iff p is a prime factor of n^2-2 and p > n. n^2-2 has at most one prime factor > n, consequently such a factor is the only prime p such that n is a solution mod p of x^2 = 2. For n such that n^2-2 has no prime factor > n (the zeros in the sequence), cf. A060515.

%H Robert Israel, <a href="/A059772/b059772.txt">Table of n, a(n) for n = 2..10000</a>

%F If n^2-2 has a (unique) prime factor p > n, then a(n) = p, else a(n) = 0.

%e a(11) = 17, since 11 is a solution mod 17 of x^2 = 2 and 11 is not a solution mod p of x^2 = 2 for primes p < 17. Although 11^2 = 2 mod 7, prime 7 is excluded because 7 < 11 and 11 = 4 mod 7.

%p f:= proc(n) local P;

%p P:= select(`>`,numtheory:-factorset(n^2-2),n);

%p if P = {} then 0 else min(P) fi

%p end proc:

%p map(f, [$2..100]); # _Robert Israel_, Feb 23 2016

%t a[n_] := Module[{P}, P = Select[FactorInteger[n^2 - 2][[All, 1]], # > n&]; If[P == {}, 0, Min[P]]];

%t Table[a[n], {n, 2, 100}] (* _Jean-François Alcover_, Apr 10 2019, from Maple *)

%Y Cf. A038873, A059770, A059771, A060515.

%K nonn

%O 2,2

%A _Klaus Brockhaus_, Feb 21 2001

%E Offset corrected by _R. J. Mathar_, Aug 21 2009