Study of dividing graph 5

The divisor graph is the non oriented graph whose vertices are the positive integers, two vertices being connected by an edge when the smallest one divides the largest one. We call chain of length l any finite sequence of pairwise distinct positive integers a (1) , a( 2) , ... , a (l) , such that, for 1 <= i <= l , a(i )and a(i +1) are connected by an edge in the divisor graph. Let f (x) denote the maximum length of the restriction of the divisor graph to the integers smaller than or equal to x . Tenenbaum has given a constructive procedure, directly transposable in form of an algorithm, which establishes the existence of a chain of integers <= x , from which the lower bound f (x) >= (c + o (1)) x/ log x is now known to follow, with c = 0, , 07 when x ->+infinity. We give here a variant of his constructive procedure that allows to get a similar lower bound with c = 0, , 37. We take advantage of the opportunity to make a large part to experimental mathematics. Moreover, we also give a lower bound of a variant of the function f (x), which will enable us to answer a question of Erd & odblac;s in a forthcoming paper.