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- W4299307103 abstract "Let $T(n)=left{begin{array}{ll}3n+1&(nhbox{ odd})frac n2&(nhbox{ even})end{array}right.$ ($ninmathbb Z$). We call the orbit of the integer $n$, the set $$ mathcal O_n:={minmathbb Z;:;exists kge0, m=T^k(n)} $$ and we put $c_i(n):=#{minmathcal O_n;:;mequiv ihbox{ mod.}18}$. Let $W$ be the set of the integers whose orbit contains $1$ and is, in the following sense, about well distributed modulo $18$ between the six elements of the set $I:={1,5,7,11,13,17}$ (the elements of {1,dots,18} that are odd and not divisible by $3$). More precisely: $$ W:=Big{ninmathbb N;:;exists kge0, T^k(n)=1hbox{ and }forall iin I, frac{c_i(n)}{sum_{iin I}c_i(n)}lefrac16+0.0215Big}. $$ We prove that $W$ has density $0$ in $mathbb N$. Consequently, if the $3x+1$ conjecture is true, most of the positive integers $n$ satisfy $$ frac{max_{iin I}c_i(n)}{sum_{iin I}c_i(n)}>frac16+0.0215. $$" @default.
- W4299307103 created "2022-10-02" @default.
- W4299307103 creator A5009110095 @default.
- W4299307103 date "2015-12-17" @default.
- W4299307103 modified "2023-09-23" @default.
- W4299307103 title "A non-uniform distribution property of most orbits, in case the $3x+1$ conjecture is true" @default.
- W4299307103 doi "https://doi.org/10.48550/arxiv.1512.05852" @default.
- W4299307103 hasPublicationYear "2015" @default.
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