A341669 Primes p such that p^7 - 1 has 8 divisors.

863, 1439, 2039, 3167, 3803, 4799, 10559, 11423, 14087, 14207, 15287, 15803, 16139, 18743, 20663, 21059, 21179, 22343, 25307, 25919, 26459, 29483, 29759, 30803, 32507, 32987, 33107, 34319, 34367, 35879, 43427, 45887, 46559, 46643, 46919, 54959, 57119, 57587

Offset: 1

Author

Jon E. Schoenfield, Feb 26 2021

Keywords

• nonn

Comments

For each term p, p^7 - 1 = (p-1)*(p^6 + p^5 + p^4 + p^3 + p^2 + p + 1) is a number of the form 2*q*r (where q and r are distinct primes): p-1 = 2*q and p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 = r.

Conjecture: sequence is infinite.

Examples

			      p =
  n   a(n)        factorization of p^7 - 1
  -  -----  ------------------------------------
  1    863  2 *  431 *        413588356833933793
  2   1439  2 *  719 *       8885189025331426081
  3   2039  2 * 1019 *      71897932302115976281
  4   3167  2 * 1583 *    1009312223899992366817
  5   3803  2 * 1901 *    3026022586778671180093
  6   4799  2 * 2399 *   12217856103420111345601
  7  10559  2 * 5279 * 1386046726502834819142721
  8  11423  2 * 5711 * 2221872233870122705845793
  9  14087  2 * 7043 * 7815232779386331437540137
		

Crossrefs

Cf. A000005, A000040, A309906, A341668.

Programs

Mathematica

Select[Range[60000], PrimeQ[#] && DivisorSigma[0, #^7 - 1] == 8 &] (* Amiram Eldar, Feb 27 2021 *)

PARI

isok(p) = isprime(p) && (numdiv(p^7-1) == 8); \\ Michel Marcus, Feb 27 2021