Non-regularity of floor(alpha + log_k(n))

TL;DR

This paper proves that the sequence floor(alpha + log_k(n)) is not k-regular when k^alpha is irrational, using novel methods that avoid automata theory, and also shows its generating function is not algebraic under certain conditions.

Contribution

It introduces a new proof technique for non-regularity of sequences related to irrational powers, bypassing automata and language theory.

Findings

Sequence floor(alpha + log_k(n)) is not k-regular if k^alpha is irrational

The generating function associated with the sequence is not algebraic when k^alpha is irrational

Provides a stronger statement on the non-regularity and non-algebraicity of the sequence and its generating function.

Abstract

This paper presents a new proof that if k^alpha is irrational then the sequence floor(alpha + log_k(n)) is not k-regular. Unlike previous proofs, the methods used do not rely on automata or language theoretic concepts. The paper also proves the stronger statement that if k^\alpha is irrational then the generating function in k non-commuting variables associated with floor(alpha + log_k(n)) is not algebraic.