OFFSET
1,1
COMMENTS
Note that there are two cases: (1) n is 2p, in which case the semiprime is Fibonacci(p)*Lucas(p) for some prime p, or (2) n is a power of a prime p^k for k > 0. In the first case, the primes p are in sequence A080327. In the second case, it appears that k=1 except for n = 8, 9 and 121. - T. D. Noe, Sep 23 2005
The associated sequence of Fibonacci numbers contains no squares, since the only Fibonacci numbers which are square are 1 and 144. Consequently this is a subsequence of A114842. - Charles R Greathouse IV, Sep 24 2012
Sequence continues as 1543?, 1709, 1741?, 1759, 1801?, 1889, 1987, ..., where ? marks uncertain terms. - Max Alekseyev, Jul 10 2016
LINKS
Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
Ron Knott, Fibonacci numbers
Blair Kelly, Fibonacci and Lucas Factorizations
EXAMPLE
a(4) = 14 because the 14th Fibonacci number 377 = 13*29 is a semiprime.
MATHEMATICA
Select[Range[200], Plus@@Last/@FactorInteger[Fibonacci[ # ]] == 2&] (Noe)
Select[Range[1500], PrimeOmega[Fibonacci[#]]==2&] (* Harvey P. Dale, Dec 13 2020 *)
PROG
(PARI) for(n=2, 9999, bigomega(fibonacci(n))==2&&print1(n", ")) \\ - M. F. Hasler, Oct 31 2012
(PARI) issemi(n)=bigomega(n)==2
is(n)=if(n%2, my(p); if(issquare(n, &p), isprime(p) && isprime(fibonacci(p)) && isprime(fibonacci(n)/fibonacci(p)), isprime(n) && issemi(fibonacci(n))), (isprime(n/2) && isprime(fibonacci(n/2)) && isprime(fibonacci(n)/fibonacci(n/2))) || n==8) \\ Charles R Greathouse IV, Oct 06 2016
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Shyam Sunder Gupta, Jul 20 2002
EXTENSIONS
More terms from Don Reble, Jul 31 2002
a(49)-a(50) from Max Alekseyev, Aug 18 2013
a(51)-a(52) from Max Alekseyev, Jul 10 2016
STATUS
approved