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A085726 Numbers n such that n-th 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 (list; graph; refs; listen; history; text; internal format)
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

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(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

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

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Last modified February 22 07:46 EST 2020. Contains 332118 sequences. (Running on oeis4.)