cp's OEIS Frontend

A065904 Integers i > 1 for which there is one prime p such that i is a solution mod p of x^4 = 2.

%I A065904 #14 Mar 18 2025 16:06:26
%S A065904 2,3,4,6,7,9,10,11,12,13,14,19,20,21,22,23,24,26,29,31,32,37,38,39,40,
%T A065904 41,42,43,44,49,50,52,53,54,59,60,61,62,64,65,70,72,73,74,75,77,79,80,
%U A065904 82,83,85,87,89,93,94,95,96,97,99,100,101,103,108,109,111,116,119,121
%N A065904 Integers i > 1 for which there is one prime p such that i is a solution mod p of x^4 = 2.
%C A065904 Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has no resp. two resp. three prime factors > i; cf. A065903 resp. A065905 resp. A065906.
%H A065904 Robert Israel, <a href="/A065904/b065904.txt">Table of n, a(n) for n = 1..10000</a>
%F A065904 a(n) = n-th integer i such that i^4 - 2 has one prime factor > i.
%e A065904 a(3) = 4, since 4 is (after 2 and 3) the third integer i for which there is one prime p > i (viz. 127) such that i is a solution mod p of x^4 = 2, or equivalently, 4^4 - 2 = 254 = 2*127 has one prime factor > 4 (cf. A065902).
%p A065904 filter:= n -> nops(select(`>`, numtheory:-factorset(n^4-2),n))=1:
%p A065904 select(filter, [$2..1000]); # _Robert Israel_, Jan 30 2017
%t A065904 okQ[n_] := Length[Select[FactorInteger[n^4 - 2][[All, 1]], # > n&]] == 1;
%t A065904 Select[Range[2, 200], okQ] (* _Jean-François Alcover_, Mar 26 2019, after _Robert Israel_ *)
%o A065904 (PARI)
%o A065904 a065904(m) = local(c,n,f,a,s,j); c = 0; n = 2; while(c<m,f = factor(n^4-2); a = matsize(f)[1]; s = []; for(j = 1,a, if(f[j,1]>n,s = concat(s,f[j,1]))); if(matsize(s)[2] == 1,print1(n,","); c++); n++)
%o A065904 a065904(70)
%Y A065904 Cf. A040028, A065902, A065903, A065905, A065906.
%K A065904 nonn
%O A065904 1,1
%A A065904 _Klaus Brockhaus_, Nov 28 2001