[seqfan] Sequences related to the fibonacci-sequence

[David Corneth]

Hi all,

I have some ideas about two existing sequences, A065108: Numbers
expressible as a product of Fibonacci numbers. and A114823: Indices of
Fibonacci numbers with 13 distinct prime factors.

So to start with A065108. There is a comment from Neil saying: "As Charles
R Greathouse IV <https://oeis.org/wiki/User:Charles_R_Greathouse_IV> recently
remarked, it would be good to have an asymptotic formula for this sequence."
Maybe this can be achieved as follows: a fibonacci-number about
phi^(e_i)/sqrt(5) where phi = (1 + sqrt(5))/2. So a product of m
fibonacci-numbers is of the form phi^(sum_(i=1..m) e_i) / sqrt(5)^m. Let
E_m = sum_(i=1..m) e_i
Now, to get an approximation of how many elements there are below some n,
we (approximately) solve phi^(E_m) / sqrt(5)^m <= n for
m = 1...ceil(log(n)/log(2)). We can do so by partitioning E_m into m parts.
Probably with e_i >= 3 so we're merely dealing with products of
fibonacci-numbers >= fibonacci(3) = 2.

I wrote the following code to do it in PARI but I'm unable to see if the
error is a conincidence or wrong.
~~~~~~~~~~~~~~~~~~~~~~~~
addhelp(countupto, "Amount of numbers below n being a product of
 fibonacci-numbers.")
countupto(n) = { my(phii = (1+sqrt(5))/2);
sum(i=1,logint(n,2),numfib(n, i))\2
}

addhelp(numfib, "Amount of numbers below n being a product of exactly m
fibonacci-numbers.")
numfib(n, m) = {my(sqrt5 = sqrt(5), phii = (1+sqrt5)/2);
n*=sqrt5^m;
qp = ceil(log(n) / log(phii));
sum(i=3*m,qp,numbpartn(i-2*m, m))
}

addhelp(numbpartn, "Number of partitions of n into m parts.")
numbpartn(n, m) = {my(p=partitions(n)); sum(i=1,#p,#p[i]==m)}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
About A114823, I'd like to add A001602 as an Xref as it may help to find
candidates k such that omega(fibonacci(k)) == 13. Also a new sequence that
lists for n the largest value for omega(fibonacci(n)) -
omega(fibonacci(n/p)) where p is in this case a prime divisor of k. Maybe
we could predict that if some k has omega(fibonacci(k))==10, a prime if it
exists, giving omega(fibonacci(p * k))==13. Could this help to find more
terms for A114823? What do you think?

Best,
David