[seqfan] Re: Observations on some odd Fibonacci numbers

[T. D. Noe]

At 5:49 PM +0000 10/2/10, Vladimir Shevelev 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?


It seems that you have found the identity

F(2n) = F(n) * L(n).

Using this recursively gives

F(2^k n) = F(n) * L(2^(k-1) n) * L(2^(k-2) n) * ... * L(n).

Best regards,

Tony