[seqfan] Re: Observations on some odd Fibonacci numbers

Alonso Del Arte alonso.delarte at gmail.com
Sat Oct 2 22:24:37 CEST 2010


My gut feeling is that this is proven (or at least stated) in Thomas Koshy's
very thorough book "Fibonacci and Lucas Numbers and Applications" (from
Wiley, 2001), a treasure trove of interesting identities for these numbers.
When I get a chance I'll see if I can find it in that book.

Al

On Sat, Oct 2, 2010 at 1:49 PM, Vladimir Shevelev <shevelev at bgu.ac.il>wrote:

>  Dear SeqFans,
>   I consider the following subsequence of Fibonacci numbers:
> 21,55,377,987,6765,17711,121393,2178309,5702887,39088169,1836311903,...
> with the definition: a(n) is the n-th odd Fibonacci number F with the
> property: F has a proper Fibonacci divisor G>1, but F/G has not.
>   I noticed (without a proof) that F/G is a Lucas number or a product of
> some Lucas numbers.
>   E.g., for F=6765, G=5 and F/G=1353=11*123; for F=2178309, G=3 and
> F/G=726103=7*47*2207; for F=1836311903, G=28657 and F/G=64079.
> Could anyone verify (or disprove) this observation for further terms of
> the sequence?
> Regards,
> Vladimir
>
>  Shevelev Vladimir‎
>
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