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%I #11 Mar 04 2021 01:42:42
%S 7,23,83,227,263,359,479,503,563,1187,2999,3803,4703,4787,4919,5939,
%T 6599,8819,10667,14159,16139,16187,18119,21227,22943,25847,26003,
%U 26903,27827,29123,29339,29663,36263,43403,44519,44963,46199,47123,48947,49103,49499
%N Primes p such that p^5 - 1 has 8 divisors.
%C 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.
%C Conjecture: sequence is infinite.
%e p = factorization
%e n a(n) p^5 - 1 of (p^5 - 1)
%e - ---- -------------- ---------------------
%e 1 7 16806 2 * 3 * 2801
%e 2 23 6436342 2 * 11 * 292561
%e 3 83 3939040642 2 * 41 * 48037081
%e 4 227 602738989906 2 * 113 * 2666986681
%e 5 263 1258284197542 2 * 131 * 4802611441
%e 6 359 5963102065798 2 * 179 * 16656709681
%e 7 479 25216079618398 2 * 239 * 52753304641
%e 8 503 32198817702742 2 * 251 * 64141071121
%e ...
%t Select[Range[50000], PrimeQ[#] && DivisorSigma[0, #^5 - 1] == 8 &] (* _Amiram Eldar_, Feb 26 2021 *)
%o (PARI) isok(p) = isprime(p) && (numdiv(p^5-1) == 8); \\ _Michel Marcus_, Feb 26 2021
%Y Cf. A000005, A000040, A309906, A341664.
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Feb 26 2021