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%I #31 Sep 08 2022 08:45:11
%S 3,10,14,20,23,26,29,32,38,43,49,56,62,64,67,68,73,76,80,83,86,89,97,
%T 107,121,128,136,137,157,164,167,172,178,197,202,211,223,229,284,293,
%U 307,311,328,373,389,397,458,487,521,541,557,577,586,619,673,857,914,929,947,1082,1151,1249,1277,1279,1306,1318,1493,1499,1667
%N Numbers n such that n-th Lucas number is a semiprime.
%C From results on the divisibility of generalized Fibonacci sequences (2nd order recurrences with various integer initial values), it follows that if n is such that n-th Lucas number is a semiprime, it is necessary but not sufficient that n have at most two distinct prime factors (A070915). That is: A000204(n) an element of A001358 implies n an element of A070915. - _Jonathan Vos Post_, Sep 22 2005
%C All numbers in this sequence have the form 2^r p^s, where p is an odd prime and r and s are not both zero. It appears that s=2 for only p=7 and 11, otherwise s=0 or 1. - _T. D. Noe_, Nov 29 2005
%C Sequence continues as 1831?, 1877?, 1901, 1951, ... where ? mark uncertain terms. - _Max Alekseyev_, Aug 18 2013
%H Blair Kelly, <a href="http://mersennus.net/fibonacci//">Fibonacci and Lucas Factorizations</a>
%t a = 1; b = 3; Do[c = a + b; If[Plus@@Last/@FactorInteger[c] == 2, Print[n]]; a = b; b = c, {n, 3, 200}] (* _Ryan Propper_, Jun 28 2005 *)
%t Select[Range[400], PrimeOmega[LucasL[#]] == 2 &] (* _Vincenzo Librandi_, Feb 12 2016 *)
%o (Magma) IsSemiprime:=func<n | &+[k[2]: k in Factorization(n)] eq 2>; [n: n in [2..300] | IsSemiprime(Lucas(n))]; // _Vincenzo Librandi_, Feb 12 2016
%o (PARI) isok(n) = bigomega(fibonacci(n+1)+fibonacci(n-1)) == 2; \\ _Michel Marcus_, Feb 12 2016
%Y Cf. A000204.
%Y Cf. A072381 (n such that Fibonacci(n) is a semiprime).
%K nonn
%O 1,1
%A _Jason Earls_, Jul 20 2003
%E More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 25 2004
%E More terms from _Ryan Propper_, Jun 28 2005
%E More terms from _T. D. Noe_, Nov 29 2005
%E a(60)-a(62) from _Max Alekseyev_, Aug 18 2013
%E a(63)-a(69) from _Sean A. Irvine_, Feb 11 2016
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