

A072381


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


21



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 ? mark uncertain terms.  Max Alekseyev, Jul 10 2016


LINKS

Table of n, a(n) for n=1..52.
Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 1720.
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

Cf. A053409, A085726 (n such that nth Lucas number is a semiprime).
Column k=2 of A303215.
Sequence in context: A154967 A271211 A341044 * A046415 A091417 A069237
Adjacent sequences: A072378 A072379 A072380 * A072382 A072383 A072384


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



