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

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, 41, 42, 43, 44, 49, 50, 52, 53, 54, 59, 60, 61, 62, 64, 65, 70, 72, 73, 74, 75, 77, 79, 80, 82, 83, 85, 87, 89, 93, 94, 95, 96, 97, 99, 100, 101, 103, 108, 109, 111, 116, 119, 121

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

Cf. A040028, A065902, A065903, A065905, A065906.

Sequence in context: A039197 A286057 A039148 * A039108 A020756 A051382

Adjacent sequences: A065901 A065902 A065903 * A065905 A065906 A065907

Keyword

nonn

Author

Klaus Brockhaus, Nov 28 2001

Status

approved