[seqfan] Observations on some odd Fibonacci numbers

[Vladimir Shevelev]

 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