This is an appeal for help. Has anyone come across the constant 2.648102…?
Here is the background, which connects with my previous posts about graphs on groups. We are interested in the clique number of the power graph of the cyclic group of order n. The generators of the group are joined to everything else; there are φ(n) of them, where φ is Euler’s function. These must be in every maximal clique. To them, we adjoin a maximum-size clique of the subgroup of order n/p, where p is the smallest prime divisor of n. Thus the size f(n) of the largest clique satisfies the recurrence f(1) = 1, f(n) = φ(n)+f(n/p), where as above p is the smallest prime divisor of n. (This was proved by Alireza, Erfanian and Abbas in 2015.)
Now, with f defined as above, I conjecture that the ratio f(n)/φ(n) is bounded, and its lim sup is the constant 2.648102….
As you might expect, the largest values of this ratio are realised by the product of the first so many primes.