cp's OEIS Frontend

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

%I A341659 #20 Apr 17 2025 18:42:00
%S A341659 59,167,383,839,1487,4259,5087,6047,6599,6719,8543,8963,9743,12227,
%T A341659 12647,13163,14087,14867,18947,20123,22643,23099,23159,24083,24239,
%U A341659 24659,25583,27107,27299,30203,30803,32507,34319,37463,37799,38603,41879,42839,44519,44687
%N A341659 Primes p such that p^3 - 1 has 8 divisors.
%C A341659 Intersection of A005385 (Safe primes p: (p-1)/2 is also prime) and A053182 (Primes p such that p^2 + p + 1 is prime).
%C A341659 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.
%C A341659 Conjecture: sequence is infinite.
%H A341659 David A. Corneth, <a href="/A341659/b341659.txt">Table of n, a(n) for n = 1..6183</a>
%e A341659      p =                    factorization
%e A341659   n  a(n)    p^3 - 1         of (p^3 - 1)
%e A341659   -  ----  ------------  -------------------
%e A341659   1    59        205378  2 *   29 *     3541
%e A341659   2   167       4657462  2 *   83 *    28057
%e A341659   3   383      56181886  2 *  191 *   147073
%e A341659   4   839     590589718  2 *  419 *   704761
%e A341659   5  1487    3288008302  2 *  743 *  2212657
%e A341659   6  4259   77254345978  2 * 2129 * 18143341
%e A341659   7  5087  131639193502  2 * 2543 * 25882657
%e A341659   8  6047  221115865822  2 * 3023 * 36572257
%e A341659   9  6599  287365339798  2 * 3299 * 43553401
%t A341659 Select[Range[50000], PrimeQ[#] && DivisorSigma[0, #^3 - 1] == 8 &] (* _Amiram Eldar_, Feb 26 2021 *)
%t A341659 Select[Prime[Range[5000]],DivisorSigma[0,#^3-1]==8&] (* _Harvey P. Dale_, Apr 17 2025 *)
%o A341659 (PARI) isok(p) = isprime(p) && (numdiv(p^3-1) == 8); \\ _Michel Marcus_, Feb 26 2021
%Y A341659 Cf. A000005, A000040, A005385, A053182, A309906.
%K A341659 nonn
%O A341659 1,1
%A A341659 _Jon E. Schoenfield_, Feb 26 2021