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A135952 Prime factors of composite Fibonacci numbers with prime indices (cf. A050937). 5
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A142100 A093838 A055604 * A039420 A043243 A044023
KEYWORD
nonn
AUTHOR
Artur Jasinski, Dec 08 2007
EXTENSIONS
Edited, corrected and extended by Max Alekseyev, Dec 12 2007
STATUS
approved

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Last modified March 1 20:05 EST 2024. Contains 370443 sequences. (Running on oeis4.)