Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4

The Terms in Lucas Sequences Divisible by Their Indices

Chris Smyth
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
James Clerk Maxwell Building
King's Buildings
Mayfield Road
Edinburgh EH9 3JZ
United Kingdom


For Lucas sequences of the first kind $ (u_n)_{n\ge 0}$ and second kind $ (v_n)_{n\ge 0}$ defined as usual by $ u_n=(\alpha ^n-\beta ^n)/(\alpha -\beta )$, $ v_n=\alpha ^n+\beta ^n$, where $ \alpha $ and $ \beta $ are either integers or conjugate quadratic integers, we describe the sets $ \{n\in\mathbb{N}:n$    divides $ u_n\}$ and $ \{n\in\mathbb{N}:n$    divides $ v_n\}$. Building on earlier work, particularly that of Somer, we show that the numbers in these sets can be written as a product of a so-called basic number, which can only be $ 1$, $ 6$ or $ 12$, and particular primes, which are described explicitly. Some properties of the set of all primes that arise in this way is also given, for each kind of sequence.

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(Concerned with sequences A006521 A014662 A016089 A023172 A057719 A091317 A129729 A140409.)

Received August 21 2009; revised version received January 29 2010. Published in Journal of Integer Sequences, January 31 2010.

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