A072381 Numbers m such that Fibonacci(m) is a semiprime.

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

Offset: 1

Author

Shyam Sunder Gupta, Jul 20 2002

Keywords

• nonn
• hard
• more

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

Examples

			a(4) = 14 because the 14th Fibonacci number 377 = 13*29 is a semiprime.
		

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

Crossrefs

Cf. A053409, A085726 (n such that n-th Lucas number is a semiprime).

Column k=2 of A303215.

Programs

Mathematica

Select[Range[200], Plus@@Last/@FactorInteger[Fibonacci[ # ]] == 2&] (Noe)
Select[Range[1500],PrimeOmega[Fibonacci[#]]==2&] (* Harvey P. Dale, Dec 13 2020 *)

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

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