A342065 - OEIS

%I #11 Feb 28 2021 19:52:19

%S 383,12227,44519,44687,56003,97523,130259,148727,160739,169007,208799,

%T 258887,270563,281783,331883,336143,353099,364979,498119,501707,

%U 550679,573107,577667,716747,753023,775367,781007,784727,861299,887543,1084247,1085159,1099139

%N Primes p such that p^9 - 1 has 16 divisors.

%C Conjecture: sequence is infinite.

%C The only primes p such that p^9 - 1 has fewer than A309906(9)=16 divisors are p=2 (2^9 - 1 = 511 = 7*73 has 4 divisors) and p=3 (3^9 - 1 = 19682 = 2*13*757 has 8 divisors).

%C For every term p, p^9 - 1 is of the form 2*q*r*s, where q = (p-1)/2, r = (p^2 + p + 1), and s = (p^6 + p^3 + 1) are primes (see Example section).

%C The Generalized Dickson's Conjecture implies there are infinitely many p such that p, (p-1)/2, p^2+p+1 and p^6+p^3+1 are prime. - _Robert Israel_, Feb 28 2021

%H Robert Israel, <a href="/A342065/b342065.txt">Table of n, a(n) for n = 1..1000</a>

%e                        factorization of p^9 - 1

%e     p =   ===================================================

%e n   a(n)  2 * (p-1)/2 * (p^2+p+1) *      (p^6 + p^3 + 1)

%e -  -----  ---------------------------------------------------

%e 1    383  2 *   191   *    147073 *          3156404483062657

%e 2  12227  2 *  6113   * 149511757 * 3341330794198073514753973

%p R:= NULL: count:= 0: q:= 1:

%p while count < 100 do

%p   q:= nextprime(q);

%p   p:= 2*q+1;

%p   if isprime(p) and isprime(p^2+p+1) and isprime(p^6+p^3+1) then

%p     count:= count+1; R:= R, p;

%p   fi

%p od:

%p R; # _Robert Israel_, Feb 28 2021

%Y Cf. A000005, A000040, A309906, A341670.

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Feb 27 2021