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

1,1

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

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.

Example

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

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 *)

Prog

(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) ];
(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

Crossrefs

Subsequence of A014574 and A136242.

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

Cf. A001157, A169635.

Sequence in context: A125536 A003763 A179853 * A190351 A267071 A219165

Adjacent sequences: A268040 A268041 A268042 * A268044 A268045 A268046

Keyword

nonn,easy

Author

Vincenzo Librandi, Jan 25 2016

Status

approved