A341666 Primes p such that p^6 - 1 has 384 divisors.

29, 43, 59, 83, 157, 193, 317, 1093, 1373, 1523, 2803, 3557, 3677, 3733, 12227, 13093, 20507, 25933, 28163, 29243, 32443, 33493, 38603, 53917, 100523, 109883, 122117, 134363, 140197, 190573, 236723, 242773, 249397, 256757, 258403, 274237, 299723, 333283

Offset: 1

Author

Jon E. Schoenfield, Feb 26 2021

Keywords

• nonn

Comments

Conjecture: sequence is infinite.

For every term p, p^6 - 1 is of the form 2^3 * 3^2 * 7 * q * r * s * t, where q, r, s, and t are distinct primes > 7, with four exceptions: p = 29, 59, 193, and 1373 (see Example section).

Examples

			  p =
n a(n)                factorization of p^6 - 1
- ---- ------------------------------------------------------
1   29 2^3 * 3^2 * 5 * 7   *  13 *     67 *     271
2   43 2^3 * 3^2     * 7   *  11 *     13 *     139 *     631
3   59 2^3 * 3^2 * 5 * 7   *  29 *    163 *    3541
4   83 2^3 * 3^2     * 7   *  19 *     41 *     367 *    2269
5  157 2^3 * 3^2     * 7   *  13 *     79 *    3499 *    8269
6  193 2^7 * 3^2     * 7   *  97 *   1783 *   37057
7  317 2^3 * 3^2     * 7   *  53 *     79 *   14401 *   33391
8 1093 2^3 * 3^2     * 7   *  13 *    547 *  398581 * 1193557
9 1373 2^3 * 3^2     * 7^3 * 229 * 627919 * 1886503
		

Crossrefs

Cf. A000005, A000040, A309906, A341657, A341667.

Programs

Mathematica

Select[Range[350000], PrimeQ[#] && DivisorSigma[0, #^6 - 1] == 384 &] (* Amiram Eldar, Feb 27 2021 *)

PARI

isok(p) = isprime(p) && (numdiv(p^6-1) == 384); \\ Michel Marcus, Feb 27 2021