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

1,1

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.

Links

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A160848 A160278 A206030 * A174929 A072430 A278126

Adjacent sequences: A385588 A385589 A385590 * A385592 A385593 A385594

Keyword

nonn

Author

Robert Israel, Aug 09 2025

Status

approved