

A072381


Numbers m such that Fibonacci(m) is a semiprime.


23



8, 9, 10, 14, 19, 22, 26, 31, 34, 41, 53, 59, 61, 71, 73, 79, 89, 94, 101, 107, 109, 113, 121, 127, 151, 167, 173, 191, 193, 199, 227, 251, 271, 277, 293, 331, 353, 397, 401, 467, 587, 599, 601, 613, 631, 653, 743, 991, 1091, 1223, 1373, 1487
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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



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



EXTENSIONS



STATUS

approved



