Several results on sequences which are similar to the positive integers

TL;DR

This paper investigates sequences similar to natural numbers based on specific properties, proving that for large indices, these sequences coincide under certain initial conditions and properties related to binary representations.

Contribution

It introduces minimal recursive sequences based on properties like divisibility by powers of two and binary digit counts, establishing their equivalence for large n under certain initial values.

Findings

Sequences with different initial terms eventually coincide for large n.

Sequences defined by binary properties stabilize to a common form.

The results apply to properties related to binary expansion and divisibility.

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 $n,\enskip x_n^{(a)}=x_n^{(4)}.$