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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 (list; graph; refs; listen; history; text; internal format)
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. 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)

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 n-th Lucas number is a semiprime).

Column k=2 of A303215.

Sequence in context: A164276 A154967 A271211 * 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

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Last modified March 29 21:32 EDT 2020. Contains 333117 sequences. (Running on oeis4.)