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%I #32 Apr 19 2024 17:37:02
%S 6,1092,1932,2730,4158,6552,11172,25998,30492,55440,76650,79632,85092,
%T 102102,150990,152082,152418,166782,211218,235662,236208,248640,
%U 264600,298410,300300,301182,317772,380310,387198,441798,476028,488418
%N Numbers k such that k^3 - 1 and k^3 + 1 are both semiprimes.
%C Obviously, k+1 and k-1 are always prime numbers.
%C 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
%H Amiram Eldar, <a href="/A268043/b268043.txt">Table of n, a(n) for n = 1..10000</a>
%H Jean-Marie De Koninck, <a href="http://ac.inf.elte.hu/Vol_021_2002/127.pdf">On the solutions of sigma_2(n) = sigma_2(n + l)</a>, Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
%e a(1) = 6 because 6^3-1 = 215 = 5*43 and 6^3+1 = 217 = 7*31, therefore 215, 217 are both semiprimes.
%t Select[Range[500000], PrimeOmega[#^3 - 1] == PrimeOmega[#^3 + 1] == 2 &]
%t Select[Range[10^6], And @@ PrimeQ[{# - 1, # + 1, #^2 - # + 1, #^2 + # + 1}] &] (* _Amiram Eldar_, Apr 19 2024 *)
%o (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) ];
%o (PARI) isok(n) = (bigomega(n^3-1) == 2) && (bigomega(n^3+1) == 2); \\ _Michel Marcus_, Jan 26 2016
%o (PARI) is(n) = isprime(n - 1) && isprime(n + 1) && isprime(n^2 - n + 1) && isprime(n^2 + n + 1); \\ _Amiram Eldar_, Apr 19 2024
%Y Subsequence of A014574 and A136242.
%Y Cf. A002384, A055494, A088707, A096173, A096175, A109373.
%Y Cf. A001157, A169635.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Jan 25 2016
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