A341665 Primes p such that p^5 - 1 has 8 divisors.

7, 23, 83, 227, 263, 359, 479, 503, 563, 1187, 2999, 3803, 4703, 4787, 4919, 5939, 6599, 8819, 10667, 14159, 16139, 16187, 18119, 21227, 22943, 25847, 26003, 26903, 27827, 29123, 29339, 29663, 36263, 43403, 44519, 44963, 46199, 47123, 48947, 49103, 49499

Offset

1,1

Comments

For each term p, p^5 - 1 = (p-1)*(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^4 + p^3 + p^2 + p + 1 = r.

Conjecture: sequence is infinite.

Links

Table of n, a(n) for n=1..41.

Example

     p =                       factorization
  n  a(n)      p^5 - 1          of (p^5 - 1)
  -  ----  --------------  ---------------------
  1     7           16806  2 *   3 *        2801
  2    23         6436342  2 *  11 *      292561
  3    83      3939040642  2 *  41 *    48037081
  4   227    602738989906  2 * 113 *  2666986681
  5   263   1258284197542  2 * 131 *  4802611441
  6   359   5963102065798  2 * 179 * 16656709681
  7   479  25216079618398  2 * 239 * 52753304641
  8   503  32198817702742  2 * 251 * 64141071121
  ...

Mathematica

Select[Range[50000], PrimeQ[#] && DivisorSigma[0, #^5 - 1] == 8 &] (* Amiram Eldar, Feb 26 2021 *)

Prog

(PARI) isok(p) = isprime(p) && (numdiv(p^5-1) == 8); \\ Michel Marcus, Feb 26 2021

Crossrefs

Cf. A000005, A000040, A309906, A341664.

Sequence in context: A145842 A086908 A093069 * A322269 A303890 A003540

Adjacent sequences: A341662 A341663 A341664 * A341666 A341667 A341668

Keyword

nonn

Author

Jon E. Schoenfield, Feb 26 2021

Status

approved