A065906 Integers i > 1 for which there are three primes p such that i is a solution mod p of x^4 = 2.

15, 48, 55, 197, 206, 221, 235, 283, 297, 408, 444, 472, 489, 577, 578, 623, 641, 677, 701, 703, 763, 854, 930, 1049, 1081, 1134, 1140, 1159, 1160, 1201, 1253, 1303, 1311, 1328, 1374, 1385, 1415, 1458, 1459, 1495, 1501, 1517, 1557, 1585, 1714, 1723, 1726

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. one resp. two prime factors > i cf. A065903 resp. A065904 resp. A065905.

Links

Table of n, a(n) for n=1..47.

Formula

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

Example

a(3) = 55, since 55 is (after 15 and 48) the third integer i for which there are three primes p > i (viz. 73, 103 and 1217) such that i is a solution mod p of x^4 = 2, or equivalently, 55^4 - 2 = 9150623 = 73*103*1217 has three prime factors > 4. (cf. A065902).

Prog

(PARI): a065906(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] == 3, print1(n, ", "); c++); n++) a065906(50)

Crossrefs

Cf. A040028, A065902, A065903, A065904, A065905.

Sequence in context: A236401 A000813 A156205 * A370912 A154060 A168305

Adjacent sequences: A065903 A065904 A065905 * A065907 A065908 A065909

Keyword

nonn

Author

Klaus Brockhaus, Nov 28 2001

Status

approved