Several results on sequences which are similar to the positive integers
| dc.creator | Shevelev, Vladimir | |
| dc.date | 2009-04-14 | |
| dc.date | 2009-04-20 | |
| dc.date.accessioned | 2026-07-07T13:05:32Z | |
| dc.date.available | 2026-07-07T13:05:32Z | |
| dc.description | 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 $n,\enskip x_n^{(a)}=x_n^{(4)}.$ | |
| dc.description | 14 pages. I did some changes in Introduction to facilitate reading of the paper | |
| dc.identifier | https://arxiv.org/abs/0904.2101 | |
| dc.identifier | http://arxiv.org/abs/0904.2101 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/227519 | |
| dc.subject | Number Theory | |
| dc.subject | 11B37 | |
| dc.title | Several results on sequences which are similar to the positive integers | |
| dc.type | text |