OFFSET
1,4
COMMENTS
The products of Fibonacci numbers are not all distinct. For example, 144 is a product of Fibonacci numbers in more than 1 way but 144 still counts as one product. 8 and 144 are the only Fibonacci numbers with this property (see A235383).
a(n+1)/a(n) seems to converge to about 1.1; up to n = 70, the value drops slightly. Maybe to sqrt(5)/2?
Fibonacci(n) ~= phi^n / sqrt(5) where phi = (1 + sqrt(5)) / 2. If m is a product of k Fibonacci numbers, m is of the form Fibonacci(n_1) *...* Fibonacci(n_k). To count the numbers just once, we restrict n_i for 1 <= i <= k.
Fibonacci(1) = Fibonacci(2) = 1 isn't counted, products with factor Fibonacci(6) = 8 aren't counted and products with the factor Fibonacci(12) = 144 aren't counted. I.e., n_i >= 3, n_i != 6 and n_i != 12.
We can write m uniquely as m = Product_{i=1..k} Fibonacci(n_i) ~= Product_{i=1..k} (phi^(n_i) / sqrt(5)) = phi^(Sum_{i=1..k} n_i) / sqrt(5)^k. To determine the number of such products up to f = Fibonacci(x) of k such Fibonacci factors, we can find an upper bound on Sum_{i=1..k} n_i of about (k*log(5)/2 + log(x)) / log(phi). This somewhat relates this sequence to the partitions.
EXAMPLE
The products of Fibonacci numbers larger than Fibonacci(7) = 13 and smaller than or equal to Fibonacci(7 + 1) = 21 are the five numbers 15, 16, 18, 20 and 21. Therefore a(7) = 5.
CROSSREFS
KEYWORD
nonn
AUTHOR
David A. Corneth, Jun 11 2017
STATUS
approved