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

1,1

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).

Links

Table of n, a(n) for n=1..38.

Example

  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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A076439 A341658 A168474 * A247896 A162357 A316741

Adjacent sequences: A341663 A341664 A341665 * A341667 A341668 A341669

Keyword

nonn

Author

Jon E. Schoenfield, Feb 26 2021

Status

approved