A268043 Numbers k such that k^3 - 1 and k^3 + 1 are both semiprimes.

6, 1092, 1932, 2730, 4158, 6552, 11172, 25998, 30492, 55440, 76650, 79632, 85092, 102102, 150990, 152082, 152418, 166782, 211218, 235662, 236208, 248640, 264600, 298410, 300300, 301182, 317772, 380310, 387198, 441798, 476028, 488418

Offset: 1

Author

Vincenzo Librandi, Jan 25 2016

Keywords

• nonn
• easy

Comments

Obviously, k+1 and k-1 are always prime numbers.

If k is a term then m = (k - 1) * (k^2 + k + 1) is a term of A169635, i.e., A001157(m) = A001157(m+2) (De Koninck, 2002). - Amiram Eldar, Apr 19 2024

Examples

			a(1) = 6 because 6^3-1 = 215 = 5*43 and 6^3+1 = 217 = 7*31, therefore 215, 217 are both semiprimes.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jean-Marie De Koninck, On the solutions of sigma_2(n) = sigma_2(n + l), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.

Crossrefs

Subsequence of A014574 and A136242.

Cf. A002384, A055494, A088707, A096173, A096175, A109373.

Cf. A001157, A169635.

Programs

Magma

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..300000] | IsSemiprime(n^3+1) and IsSemiprime(n^3-1) ];

Mathematica

Select[Range[500000], PrimeOmega[#^3 - 1] == PrimeOmega[#^3 + 1] == 2 &]
Select[Range[10^6], And @@ PrimeQ[{# - 1, # + 1, #^2 - # + 1, #^2 + # + 1}] &] (* Amiram Eldar, Apr 19 2024 *)

PARI

isok(n) = (bigomega(n^3-1) == 2) && (bigomega(n^3+1) == 2); \\ Michel Marcus, Jan 26 2016

PARI

is(n) = isprime(n - 1) && isprime(n + 1) && isprime(n^2 - n + 1) && isprime(n^2 + n + 1); \\ Amiram Eldar, Apr 19 2024