A065905 - OEIS

%I #2 Mar 30 2012 17:27:34

%S 5,8,16,17,18,25,27,28,30,33,34,35,36,45,46,47,51,56,57,58,63,66,67,

%T 68,69,71,76,78,81,84,86,88,90,91,92,98,102,104,105,106,107,110,112,

%U 113,114,115,117,118,120,122,123,125,126,127,131,132,133,134,135,136,137

%N Integers i > 1 for which there are two primes p such that i is a solution mod p of x^4 = 2.

%C 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. one resp. three prime factors > i cf. A065903 resp. A065904 resp. A065906.

%F a(n) = n-th integer i such that i^4 - 2 has two prime factors > i.

%e a(3) = 16, since 16 is (after 5 and 8) the third integer i for which there are two primes p > i (viz. 31 and 151) such that i is a solution mod p of x^4 = 2, or equivalently, 16^4 - 2 = 65534 = 2*7*31*151 has two prime factors > 4. (cf. A065902).

%o (PARI): a065905(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] == 2,print1(n,","); c++); n++) a065905(65)

%Y Cf. A040028, A065902, A065903, A065904, A065906.

%K nonn

%O 1,1

%A _Klaus Brockhaus_, Nov 28 2001