OFFSET
1,1
COMMENTS
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n-th integer i such that i^4 - 2 has one prime factor > i.
EXAMPLE
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).
MAPLE
filter:= n -> nops(select(`>`, numtheory:-factorset(n^4-2), n))=1:
select(filter, [$2..1000]); # Robert Israel, Jan 30 2017
MATHEMATICA
okQ[n_] := Length[Select[FactorInteger[n^4 - 2][[All, 1]], # > n&]] == 1;
Select[Range[2, 200], okQ] (* Jean-François Alcover, Mar 26 2019, after Robert Israel *)
PROG
(PARI): 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++) a065904(70)
CROSSREFS
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Nov 28 2001
STATUS
approved