相关论文: Automaticity of the sequence of the last nonzero d…

In this paper we study the sequence of strings of $k$ last nonzero digits of $n!$ in a given base $b$. We determine for which $b$ this sequence is automatic and show how to generate it using a uniform morphism. We also compute how often…

Cobham's theorem asserts that if a sequence is automatic with respect to two multiplicatively independent bases, then it is ultimately periodic. We prove a stronger density version of the result: if two sequences which are automatic with…

In this paper we study the sequences defined by the last and the last non-zero digits of $n^n$ in base $b$. For the sequence given by the last digits of $n^n$ in base $b$, we prove its periodicity using different techniques than those used…

The third-named author recently proved [Israel J. of Math. 258 (2023), 475--502] that there are infinitely many \textit{collisions} of the base-2 and base-3 sum-of-digits functions. In other words, the equation \[ s_2(n)=s_3(n) \] admits…

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration…

If we form a decimal where the nth digit is the last non-zero digit of $n!$ (likewise, the last non-zero digit of $n^n$), we obtain an irrational number

Understanding the distribution of digits in the expansions of perfect powers in different bases is difficult. Rather than consider the asymptotic digit distributions, we consider the base-10 digits of a restricted sequence of powers of two.…

We make a start on one of George McNulty's Dozen Easy Problems: "Which finite automatic algebras are dualizable?" We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra…

We prove that the sequence of the last nonzero digits of factorials in every integer base $b>2$ is not eventually periodic. We also extend the Adamczewski--Bugeaud criterion, originally formulated for integer base expansions, to Cantor base…

The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.

All the already known results on self descriptive numbers, together with the demonstration of the uniqueness for bases greater than 6, are here obtained through a systematic scheme of proof and not trial and error. The proof is also…

We discuss the use of negative bases in automatic sequences. Recently the theorem-prover Walnut has been extended to allow the use of base (-k) to express variables, thus permitting quantification over Z instead of N. This enables us to…

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…

Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the…

Zero automata are a probabilistic extension of parity automata on infinite trees. The satisfiability of a certain probabilistic variant of mso, called tmso + zero, reduces to the emptiness problem for zero automata. We introduce a variant…

The comma sequence (1, 12, 35, 94, ...) is the lexicographically earliest sequence such that the difference of consecutive terms equals the concatenation of the digits on either side of the comma separating them. The behavior of a…

Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with…

Denote by $\lambda(n)$ Liouville's function concerning the parity of the number of prime divisors of $n$. Using a theorem of Allouche, Mend\`es France, and Peyri\`ere and many classical results from the theory of the distribution of prime…

In formal languages and automata theory, the magic number problem can be formulated as follows: for a given integer n, is it possible to find a number d in the range [n,2^n] such that there is no minimal deterministic finite automaton with…

We show that any automatic multiplicative sequence either coincides with a Dirichlet character or is identically zero when restricted to integers not divisible by small primes. This answers a question of Bell, Bruin and Coons. A similar…