A136242 -id:A136242 - cp's OEIS frontend

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

Vincenzo Librandi, Jan 25 2016

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

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

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.

Subsequence of A014574 and A136242.
Cf. A002384, A055494, A088707, A096173, A096175, A109373.
Cf. A001157, A169635.

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) ];

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 *)

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

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

A136243 Numbers k in A008864 such that k^2 - k - 1 is prime.

Original entry on oeis.org

3, 4, 6, 12, 14, 20, 32, 42, 54, 60, 84, 90, 102, 104, 132, 150, 164, 182, 192, 194, 200, 234, 242, 264, 282, 332, 350, 374, 402, 420, 432, 434, 450, 462, 464, 500, 542, 570, 572, 660, 674, 684, 692, 710, 740, 744, 762, 770, 810, 864, 882, 942, 1014, 1040

Offset: 1

Lekraj Beedassy, Dec 24 2007

nonn

See A053185 for the primes associated with a(n).

Cf. A008864, A053184, A136242.

isok(k) = isprime(k-1) && isprime(k^2-k-1); \\ Michel Marcus, Dec 19 2022

a(n) = A053184(n) + 1.

a(20)=194 inserted by Georg Fischer, Dec 18 2022

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

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A136243 Numbers k in A008864 such that k^2 - k - 1 is prime.

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