On quotients of values of Euler's function on the Catalan numbers
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- 作者：Bojan Ba&scaron ; ić bojan.basic@dmi.uns.ac.rs" class="auth_mail" title="E-mail the corresponding author
- 关键词：11A25 ; 11N32 ; 11N37 ; 11B65 ; 11A63
- 刊名：Journal of Number Theory
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：169
- 期：Complete
- 页码：160-173
- 全文大小：358 K
文摘
In a recent work, Luca and Stănică examined quotients of the form 
, where φ is Euler's totient function and C0,C1,C2… is the sequence of the Catalan numbers. They observed that the number 4 (and analogously ) appears noticeably often as a value of these quotients. We give an explanation of this phenomenon, based on Dickson's conjecture. It turns out not only that the value 4 is (in a certain sense) special in relation to the quotients , but also that the value 4k has similar “special” properties with respect to the quotients , and in particular we show that Dickson's conjecture implies that, for each k , the number 4k appears infinitely often as a value of the quotients .
, where φ is Euler's totient function and C0,C1,C2… is the sequence of the Catalan numbers. They observed that the number 4 (and analogously ) appears noticeably often as a value of these quotients. We give an explanation of this phenomenon, based on Dickson's conjecture. It turns out not only that the value 4 is (in a certain sense) special in relation to the quotients , but also that the value 4k has similar “special” properties with respect to the quotients , and in particular we show that Dickson's conjecture implies that, for each k , the number 4k appears infinitely often as a value of the quotients .