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A385591 Numbers k such that both k^3 - 1 and k^3 + 1 are triprimes.

%I A385591 #31 Aug 11 2025 01:24:20
%S A385591 66,132,180,228,240,288,294,336,378,420,462,600,612,660,678,702,882,
%T A385591 918,960,1116,1164,1278,1302,1320,1800,2550,2562,3270,3300,3372,3408,
%U A385591 3438,3822,3882,3990,4050,4422,4536,4812,5040,5088,5208,5250,5418,5748,5754,5778,5838,6882,6960,7128,7182,7254
%N A385591 Numbers k such that both k^3 - 1 and k^3 + 1 are triprimes.
%C A385591 Numbers k such that k^3 - 1 and k^3 + 1 each have 3 prime factors, counted with multiplicity.
%C A385591 All terms are divisible by 6.
%C A385591 The Generalized Bunyakovsky Conjecture implies there are infinitely many j such that 6+7*j, 32 + 35*j, 1225 * j^2 + 2205 * j + 993 and 175 * j^2 + 305 * j + 133 are all prime.  For such j, 31 + 35*j is a term of the sequence.  Thus the conjecture implies the sequence is infinite. The first two such j are 1 and 31, corresponding to a(1) = 66 and a(20) = 1116.
%H A385591 Robert Israel, <a href="/A385591/b385591.txt">Table of n, a(n) for n = 1..10000</a>
%e A385591 a(3) = 180 is a term because 180^3 - 1 = 5831999 = 31 * 179 * 1051 and 5832001 = 7 * 181 * 4603 are each products of 3 primes.
%p A385591 filter:= k -> numtheory:-bigomega(k-1) + numtheory:-bigomega(k^2 + k + 1) = 3 and
%p A385591 numtheory:-bigomega(k+1) + numtheory:-bigomega(k^2 - k + 1) = 3:
%p A385591 select(filter, [seq(i,i=6 .. 10000, 6);
%t A385591 Select[Range[7500], PrimeOmega[#^3 - 1] == PrimeOmega[#^3 + 1] == 3 &] (* _Amiram Eldar_, Aug 10 2025 *)
%Y A385591 Cf.  A001093, A068601, A014612. Intersection of A115403 and A386915.
%K A385591 nonn
%O A385591 1,1
%A A385591 _Robert Israel_, Aug 09 2025