

A085726


Numbers n such that nth Lucas number is a semiprime.


3



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
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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 nth 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


LINKS

Table of n, a(n) for n=1..69.
Blair Kelly, Fibonacci and Lucas Factorizations


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(n1)) == 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
Adjacent sequences: A085723 A085724 A085725 * A085727 A085728 A085729


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



