[seqfan] Re: Observations on some odd Fibonacci numbers

Richard Mathar mathar at strw.leidenuniv.nl
Tue Oct 5 18:27:03 CEST 2010

http://list.seqfan.eu/pipermail/seqfan/2010-October/006140.html

vs>    I consider the following subsequence of Fibonacci numbers:
vs> 21,55,377,987,6765,17711,121393,2178309,5702887,39088169,1836311903,...
vs> with the definition: a(n) is the n-th odd Fibonacci number F with the
vs> property: F has a proper Fibonacci divisor G>1, but F/G has not.

Note also that 6765 is not in my list because 6765 = 3*2255
where 2255 has a proper Fibonacci divisor (that is, 55). So my interpretation
of the definition is that no odd F with two proper divisors in A000045
are in a(n). I am not sure whether to admit cases where F has proper
Fibonacci divisors G of both types.

vs> I noticed (without a proof) that F/G is a Lucas number or a product of
vs> some Lucas numbers.

The examples
n= 49, F=7778742049, G=13, F/G=598364773, Lprod=false
n= 98, F=135301852344706746049, G=13, F/G=10407834795746672773, Lprod=false
n= 121, F=8670007398507948658051921, G=89, F/G=97415813466381445596089, Lprod=false
n= 169, F=93202207781383214849429075266681969, G=233, F/G=400009475456580321242184872389193, Lprod=false

are the first counter-examples according to my calculation which are in the
list of these a(n) where F/G is not a Lucas number or product of such.

http://mersennus.net/fibonacci/f1000.txt
J Brillhart et al, "Tables of Fibonacci and Lucas Factorizations" Math Comp 50 (1988) 252, http://www.jstor.org/stable/2007928