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

66, 132, 180, 228, 240, 288, 294, 336, 378, 420, 462, 600, 612, 660, 678, 702, 882, 918, 960, 1116, 1164, 1278, 1302, 1320, 1800, 2550, 2562, 3270, 3300, 3372, 3408, 3438, 3822, 3882, 3990, 4050, 4422, 4536, 4812, 5040, 5088, 5208, 5250, 5418, 5748, 5754, 5778, 5838, 6882, 6960, 7128, 7182, 7254

Offset: 1

Author

Robert Israel, Aug 09 2025

Keywords

• nonn

Comments

Numbers k such that k^3 - 1 and k^3 + 1 each have 3 prime factors, counted with multiplicity.

All terms are divisible by 6.

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001093, A068601, A014612. Intersection of A115403 and A386915.

Programs

Maple

filter:= k -> numtheory:-bigomega(k-1) + numtheory:-bigomega(k^2 + k + 1) = 3 and
numtheory:-bigomega(k+1) + numtheory:-bigomega(k^2 - k + 1) = 3:
select(filter, [seq(i,i=6 .. 10000, 6);

Mathematica

Select[Range[7500], PrimeOmega[#^3 - 1] == PrimeOmega[#^3 + 1] == 3 &] (* Amiram Eldar, Aug 10 2025 *)