A341659 Primes p such that p^3 - 1 has 8 divisors.

59, 167, 383, 839, 1487, 4259, 5087, 6047, 6599, 6719, 8543, 8963, 9743, 12227, 12647, 13163, 14087, 14867, 18947, 20123, 22643, 23099, 23159, 24083, 24239, 24659, 25583, 27107, 27299, 30203, 30803, 32507, 34319, 37463, 37799, 38603, 41879, 42839, 44519, 44687

Offset

1,1

Comments

Intersection of A005385 (Safe primes p: (p-1)/2 is also prime) and A053182 (Primes p such that p^2 + p + 1 is prime).

For each term p, p^3 - 1 = (p-1)*(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^2 + p + 1 = r.

Conjecture: sequence is infinite.

Links

David A. Corneth, Table of n, a(n) for n = 1..6183

Example

     p =                    factorization
  n  a(n)    p^3 - 1         of (p^3 - 1)
  -  ----  ------------  -------------------
  1    59        205378  2 *   29 *     3541
  2   167       4657462  2 *   83 *    28057
  3   383      56181886  2 *  191 *   147073
  4   839     590589718  2 *  419 *   704761
  5  1487    3288008302  2 *  743 *  2212657
  6  4259   77254345978  2 * 2129 * 18143341
  7  5087  131639193502  2 * 2543 * 25882657
  8  6047  221115865822  2 * 3023 * 36572257
  9  6599  287365339798  2 * 3299 * 43553401

Mathematica

Select[Range[50000], PrimeQ[#] && DivisorSigma[0, #^3 - 1] == 8 &] (* Amiram Eldar, Feb 26 2021 *)

Prog

(PARI) isok(p) = isprime(p) && (numdiv(p^3-1) == 8); \\ Michel Marcus, Feb 26 2021

Crossrefs

Cf. A000005, A000040, A005385, A053182, A309906.

Sequence in context: A255352 A044391 A044772 * A279099 A142799 A250025

Adjacent sequences: A341656 A341657 A341658 * A341660 A341661 A341662

Keyword

nonn

Author

Jon E. Schoenfield, Feb 26 2021

Status

approved