This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386915 #17 Aug 08 2025 09:58:10 %S A386915 4,5,15,27,32,42,44,48,50,59,60,66,72,75,78,84,98,104,108,114,119,132, %T A386915 140,143,147,152,162,167,174,180,182,188,200,203,206,212,215,218,224, %U A386915 228,234,236,240,252,258,264,266,270,279,288,290,294,308,318,336,338,342,350,374,378,383,384,390 %N A386915 Numbers k such that k^3 - 1 is a triprime. %C A386915 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. %C A386915 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. %C A386915 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. %C A386915 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)). %H A386915 Robert Israel, <a href="/A386915/b386915.txt">Table of n, a(n) for n = 1..10000</a> %e A386915 a(3) = 15 is a term because 15^3 - 1 = 3374 = 2 * 7 * 241 is the product of three primes. %p A386915 select(t -> numtheory:-bigomega(t^3-1)=3, [$1..1000]); %t A386915 Select[Range[400], PrimeOmega[#^3 - 1] == 3 &] (* _Amiram Eldar_, Aug 08 2025 *) %o A386915 (PARI) isok(k) = bigomega(k^3-1) == 3; \\ _Michel Marcus_, Aug 08 2025 %Y A386915 Cf. A001358, A014612, A068601, A115403. %K A386915 nonn %O A386915 1,1 %A A386915 _Robert Israel_, Aug 07 2025