A135952 Prime factors of composite Fibonacci numbers with prime indices (cf. A050937).

37, 73, 113, 149, 157, 193, 269, 277, 313, 353, 389, 397, 457, 557, 613, 673, 677, 733, 757, 877, 953, 977, 997, 1069, 1093, 1153, 1213, 1237, 1453, 1657, 1753, 1873, 1877, 1933, 1949, 1993, 2017, 2137, 2221, 2237, 2309, 2333, 2417, 2473, 2557, 2593, 2749, 2777, 2789, 2797, 2857, 2909, 2917, 3217, 3253, 3313, 3517, 3557, 3733, 4013, 4057, 4177, 4273, 4349, 4357, 4513, 4637, 4733, 4909, 4933

Offset

1,1

Comments

All numbers in this sequence are congruent to 1 mod 4. - Max Alekseyev.

If Fibonacci(n) is divisible by a prime p of the form 4k+3 then n is even. To prove this statement it is enough to show that (1+sqrt(5))/(1-sqrt(5)) is never a square modulo such p (which is a straightforward exercise).

The n-th prime p is an element of this sequence iff A001602(n) is prime and A051694(n)=A000045(A001602(n))>p. - Max Alekseyev

Links

Hans Havermann, Table of n, a(n) for n = 1..5000

Mathematica

a = {}; k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], s = FactorInteger[Fibonacci[Prime[n]]]; c = Length[s]; Do[AppendTo[k, s[[m]][[1]]], {m, 1, c}]], {n, 2, 60}]; Union[k]

Crossrefs

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852.

Sequence in context: A142100 A093838 A055604 * A039420 A043243 A044023

Adjacent sequences: A135949 A135950 A135951 * A135953 A135954 A135955

Keyword

nonn

Author

Artur Jasinski, Dec 08 2007

Extensions

Edited, corrected and extended by Max Alekseyev, Dec 12 2007

Status

approved