 

Abhishek Jha
On terms in a dynamical divisibility sequence having a fixed G.C.D with their indices
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Published: 
August 22, 2022. 
Keywords: 
asymptotic density; Divisibility sequence; greatest common divisor; arithmetic dynamics; dynamical sequence; polynomial map. 
Subject [2010]: 
Primary: 11C08. Secondary: 11A05, 11B05. 


Abstract
Let F and G be integer polynomials where F has degree at least 2. Define the sequence (a_{n}) by a_{n}=F(a_{n1}) and a_{0}=0. Let B_{F,G,k} be the set of all positive integers n
such that k  gcd(G(n),a_{n}) and if p  gcd(G(n),a_{n}) for some p, then p  k.
Let A_{F,G,k} be the subset of B_{F,G,k} such that A_{F,G,k}={n ≥ 1 :
gcd(G(n),a_{n})=k}. In this article, we prove that the asymptotic density of A_{F,G,k}
and B_{F,G,k} exists for a class of (F,G) and also compute the explicit density of A_{F,G,k}
and B_{F,G,k} for G(x)=x.


Acknowledgements
I would like to thank Emanuele Tron and Seoyoung Kim for looking at the article and providing valuable comments to improve its quality. I am thankful to Peter Mueller, David Speyer and Will Sawin for their answers on MathOverflow post [7] and Thomas Tucker for helpful discussions regarding Proposition 2.2 of the paper. I am grateful to Ayan Nath for his constant support and helpful advice. I am indebted to the anonymous referee for helpful comments.


Author information
Abhishek Jha:
Indraprastha Institute of Information Technology
New Delhi, India
abhishek20553@iiitd.ac.in

