A065905 Integers i > 1 for which there are two primes p such that i is a solution mod p of x^4 = 2.

5, 8, 16, 17, 18, 25, 27, 28, 30, 33, 34, 35, 36, 45, 46, 47, 51, 56, 57, 58, 63, 66, 67, 68, 69, 71, 76, 78, 81, 84, 86, 88, 90, 91, 92, 98, 102, 104, 105, 106, 107, 110, 112, 113, 114, 115, 117, 118, 120, 122, 123, 125, 126, 127, 131, 132, 133, 134, 135, 136, 137

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. three prime factors > i cf. A065903 resp. A065904 resp. A065906.

Links

Table of n, a(n) for n=1..61.

Formula

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

Example

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).

Prog

(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)

Crossrefs

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

Sequence in context: A141536 A314561 A314562 * A286056 A314563 A314564

Adjacent sequences: A065902 A065903 A065904 * A065906 A065907 A065908

Keyword

nonn

Author

Klaus Brockhaus, Nov 28 2001

Status

approved