  Investigation of divisibility properties of natural numbers is one of the
most important themes in the theory of numbers. Various tools have been
developed over the centuries to discover and study the various patterns in the
sequence of natural numbers in the context of divisibility. In the present
paper, we study the divisibility of natural numbers using the framework of a
growing complex network. In particular, using tools from the field of
statistical inference, we show that the network is scale-free but has a
non-stationary degree distribution. Along with this, we report a new kind of
similarity pattern for the local clustering, which we call "stretching
similarity", in this network. We also show that the various characteristics
like average degree, global clustering coefficient and assortativity
coefficient of the network vary smoothly with the size of the network. Using
analytical arguments we estimate the asymptotic behavior of global clustering
and average degree which is validated using numerical analysis.
