Several results on sequences which are similar to the positive integers

Abstract

Sequence of positive integers $\{x_n\}_{n\geq1}$ is called similar to $\mathbb {N}$ respectively a given property $A$ if for every $n\geq1$ the numbers $x_n$ and $n$ are in the same class of equivalence respectively $A\enskip(x_n\sim n (prop \enskip A).$ If $x_1=a(>1)\sim1 (prop\enskip A)$ and $x_n>x_{n-1}$ with the condition that $x_n$ is the nearest to $x_{n-1}$ number such that $x_n\sim n (prop \enskip A),$ then the sequence $\{x_n\}$ is called minimal recursive with the first term $a\enskip(\{x_n^{(a)}\}).$ 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 $n, \enskip x_n^{(a)}=x_n^{(3)};$ in the second case we prove that, for $a>4$ and sufficiently large $n,\enskip x_n^{(a)}=x_n^{(4)}.$