  We partition a series of natural numbers into infinite number sequences. We
consider two partitioning options: (a) a forest of unary trees with recurrence
formula of Mersenne numbers, and (b) a set of arithmetic progressions with
difference $2^k$. Every tree starts with an even number, and any even number
starts a certain tree. In the partitioning into arithmetic progressions, each
progression starts with a Mersenne number, and each Mersenne number is the
beginning of a particular arithmetic progression. Unary trees starting from
some term are contained in OEIS A036991 (compact Dyck path codes), so we
consider A036991 as a backbone of the partitions. In particular, we prove the
existence of an arithmetic progression of any length in A036991. As a result of
the partitions, we obtain a Natural Matrix with a packing function that
captures the bijection between the set of natural numbers and the set of
ordered pairs of natural numbers. In Natural Matrix, the even natural numbers
are located on the $x$-axis, so the selection of primes in the considered
arithmetic progressions is greatly simplified. A method for proving the
infinity of primes in A036991 is proposed. In this regard, an attempt is made
to reduce the Linnik's constant to $2$.
