A115403 -id:A115403 - cp's OEIS frontend

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

Robert Israel, Aug 09 2025

nonn

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.

			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.

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

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

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);

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

A386915 Numbers k such that k^3 - 1 is a triprime.

Original entry on oeis.org

4, 5, 15, 27, 32, 42, 44, 48, 50, 59, 60, 66, 72, 75, 78, 84, 98, 104, 108, 114, 119, 132, 140, 143, 147, 152, 162, 167, 174, 180, 182, 188, 200, 203, 206, 212, 215, 218, 224, 228, 234, 236, 240, 252, 258, 264, 266, 270, 279, 288, 290, 294, 308, 318, 336, 338, 342, 350, 374, 378, 383, 384, 390

Offset: 1

Robert Israel, Aug 07 2025

nonn

Numbers k such that either k-1 is prime and k^2 + k + 1 is a semiprime, or k-1 is a semiprime and k^2 + k + 1 is prime.
If k is odd, k-1 = 2*p for a prime p such that 4*p^2 + 6*p + 3 is prime.  The Generalized Bunyakovsky conjecture implies that there are infinitely many of these.
The Generalized Bunyakovsky conjecture also implies that there are infinitely many j such that 14*j + 3, 28*j^2 + 18*j + 3, 7*j + 2 and 196*j^2 + 154*j + 31 are all prime.  This implies that both k = 14*j + 4 and k + 1 are terms of the sequence.
There are no k where k, k + 1 and k + 2 are all terms of the sequence, since there are no terms == 1 (mod 3) except 4 (if k == 1 (mod 3), then k^3 == 1 (mod 9)).

			a(3) = 15 is a term because 15^3 - 1 = 3374 = 2 * 7 * 241 is the product of three primes.

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

Cf. A001358, A014612, A068601, A115403.

select(t -> numtheory:-bigomega(t^3-1)=3, [$1..1000]);

Select[Range[400], PrimeOmega[#^3 - 1] == 3 &] (* Amiram Eldar, Aug 08 2025 *)

isok(k) = bigomega(k^3-1) == 3; \\ Michel Marcus, Aug 08 2025

A115402 Difference between 3-almostprime(n) and 3-almostprime(n+3).

Original entry on oeis.org

12, 15, 10, 10, 15, 16, 15, 8, 8, 18, 16, 16, 7, 9, 8, 8, 17, 22, 21, 10, 7, 11, 12, 11, 7, 10, 9, 13, 14, 22, 18, 15, 7, 16, 12, 16, 7, 7, 4, 4, 10, 12, 13, 8, 9, 19, 22, 27, 23, 19, 14, 8, 11, 8

Offset: 1

Jonathan Vos Post, Jan 22 2006

			a(1) = A014612(1+3) - A014612(1) = 20 - 8 = 12.
a(2) = A014612(2+3) - A014612(2) = 27 - 12 = 15.
a(3) = A014612(3+3) - A014612(3) = 28 - 18 = 10.
a(39) = A014612(39+3) - A014612(39) = 174 - 170 = 4.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A014612, A056809, A067813, A113789, A115403.

Last[#]-First[#]&/@Partition[Select[Range[300],PrimeOmega[#]==3&],4,1] (* Harvey P. Dale, Nov 09 2012 *)

a(n) = A014612(n+3) - A014612(n).

A119956 Numbers n such that n^3+1=pqr where p,q,r are distinct primes.

Original entry on oeis.org

9, 10, 12, 13, 21, 25, 30, 34, 36, 40, 46, 52, 66, 76, 81, 90, 96, 118, 126, 130, 132, 142, 144, 154, 165, 172, 177, 180, 193, 196, 198, 204, 216, 226, 228, 238, 240, 246, 250, 256, 262, 268, 273, 282, 294, 312, 333, 336, 345, 346, 366, 370, 372, 378, 393, 400

Offset: 1

James R. Buddenhagen, Aug 02 2006

nonn

A115403 is a supersequence not requiring that p,q,r are distinct.

			9^3+1=2*5*73 a product of 3 distinct primes, so 9 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A115403.

Select[Range[400], Last/@FactorInteger[#^3 + 1] == {1, 1, 1}&] (* Vincenzo Librandi, Sep 15 2016 *)

cp's OEIS Frontend

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

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A386915 Numbers k such that k^3 - 1 is a triprime.

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A115402 Difference between 3-almostprime(n) and 3-almostprime(n+3).

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A119956 Numbers n such that n^3+1=p*q*r where p,q,r are distinct primes.

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A119956 Numbers n such that n^3+1=pqr where p,q,r are distinct primes.