

A065904


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


5



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
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OFFSET

1,1


COMMENTS

Solutions mod p are represented by integers from 0 to p1. 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) = nth 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^42), 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] (* JeanFrançois 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^42); 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



