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A085726
Numbers n such that n-th Lucas number is a semiprime.
4
3, 10, 14, 20, 23, 26, 29, 32, 38, 43, 49, 56, 62, 64, 67, 68, 73, 76, 80, 83, 86, 89, 97, 107, 121, 128, 136, 137, 157, 164, 167, 172, 178, 197, 202, 211, 223, 229, 284, 293, 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
OFFSET
1,1
COMMENTS
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
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
Sequence continues as 1831?, 1877?, 1901, 1951, ... where ? mark uncertain terms. - Max Alekseyev, Aug 18 2013
MATHEMATICA
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 *)
Select[Range[400], PrimeOmega[LucasL[#]] == 2 &] (* Vincenzo Librandi, Feb 12 2016 *)
PROG
(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
(PARI) isok(n) = bigomega(fibonacci(n+1)+fibonacci(n-1)) == 2; \\ Michel Marcus, Feb 12 2016
CROSSREFS
Cf. A000204.
Cf. A072381 (n such that Fibonacci(n) is a semiprime).
Sequence in context: A146309 A288169 A283867 * A287115 A063796 A063221
KEYWORD
nonn
AUTHOR
Jason Earls, Jul 20 2003
EXTENSIONS
More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 25 2004
More terms from Ryan Propper, Jun 28 2005
More terms from T. D. Noe, Nov 29 2005
a(60)-a(62) from Max Alekseyev, Aug 18 2013
a(63)-a(69) from Sean A. Irvine, Feb 11 2016
STATUS
approved