Abstract
AbstractLet $$b \ge 2$$
b
≥
2
be an integer base with prime factors $$p_1, \ldots , p_s$$
p
1
,
…
,
p
s
. In this paper we study sequences of “b-adic valuations” and last nonzero digits in b-adic expansions of the values $$f(n) = (f_1(n), \ldots , f_s(n))$$
f
(
n
)
=
(
f
1
(
n
)
,
…
,
f
s
(
n
)
)
, where each $$f_i$$
f
i
is a $$p_i$$
p
i
-adic analytic function. We give a complete classification concerning k-regularity of these sequences, which generalizes a result obtained for b prime by Shu and Yao. As an application, we strengthen a theorem by Murru and Sanna on b-adic valuations of Lucas sequences of the first kind. Moreover, we derive a method to determine precisely which terms of these sequences can be represented as a sum of three squares.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference42 articles.
1. Allouche, J.-P., Shallit, J.: The ring of $$k$$-regular sequences. Theoret. Comput. Sci. 98(2), 163–197 (1992)
2. Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)
3. Allouche, J.-P., Shallit, J.: The ring of $$k$$-regular sequences II. Theoret. Comput. Sci. 307(1), 3–29 (2003)
4. Bell, J.P.: A generalization of Cobham’s theorem for regular sequences, Sém. Lothar. Comb. 54A, Art. B54Ap, 15 (2005/07)
5. Bell, J.P.: $$p$$-Adic valuations and $$k$$-regular sequences. Discret. Math. 307(23), 3070–3075 (2007)