[seqfan] Products of Fibonacci numbers

[Charles Greathouse]

Fellow sequence fans, I found myself wondering about the asymptotics
of products of Fibonacci numbers, A065108. I was lead to post
http://mathoverflow.net/questions/102540
where I was pointed to the work of Luca, Pomerance, Porubský, and
Wagner. In particular, they consider the *group* generated by the
Fibonacci numbers, which gives rise to A178772. My question, then,
corresponds to the simpler issue of the *semigroup* generated by the
Fibonacci numbers.

Ignoring F_1 = F_2 = 1, F_6 = 8, and F_12 = 144, a number is a product
of Fibonacci numbers (if at all) in a unique way. Consequently, the
problem reduces to one of counting partitions. Pomerance discusses
this here:
http://www.math.dartmouth.edu/~carlp/rademacherlecture3.pdf (pages 8-13)
where a sketch is drawn up for an asymptotic formula, approximately
40.3559...^sqrt(log(x))
if I compute correctly.

But I now find myself about to leave on a trip and unable to work on
it for at least two weeks. Is anyone interested in working on this
beautiful problem? It would be wonderful to have a formula rather than
a simple ansatz, and indeed Dr. Sloane asks for one at A065108. I can
promise nothing but my appreciation and the eternal glory of credit on
that sequence.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University