Several results on sequences which are similar to the positive integers

Abstract

Sequence of positive integers $\{x_n{\}}_{n\ge 1}$ is called similar to $\mathbb{N}$ respectively a given property $A$ if for every $n\ge 1$ the numbers $x_n$ and $n$ are in the same class of equivalence respectively $A(x_n\sim n(propA).$ If $x_1=a(>1)\sim 1(propA)$ and $x_n>x_{n-1}$ with the condition that $x_n$ is the nearest to $x_{n-1}$ number such that $x_n\sim n(propA),$ then the sequence $\{x_n\}$ is called minimal recursive with the first term $a(\{x_n^{(a)}\}).$ We study two cases: $A=A_1$ is the value of exponent of the highest power of 2 dividing an integer and $A=A_2$ is the parity of the number of ones in the binary expansion of an integer. In the first case we prove that, for sufficiently large $n,x_n^{(a)}=x_n^{(3)};$ in the second case we prove that, for $a>4$ and sufficiently large $n,x_n^{(a)}=x_n^{(4)}.$

arXiv comment14 pages. I did some changes in Introduction to facilitate reading of the paper

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