On a Luschny question

TL;DR

This paper confirms a conjecture relating Euler polynomial denominators to 2-adic valuations and characteristic sequences, answering a question posed by Peter Luschny.

Contribution

It provides a proof that the denominator of the difference of Euler polynomials at x and 1 equals a specific power of 2, linking it to 2-adic valuations and characteristic sequences.

Findings

Confirmed the conjecture relating Euler polynomial denominators to 2-adic valuations.

Established the equivalence between the denominator formula and Luschny's question.

Connected the problem to characteristic sequences and 2-adic order properties.

Abstract

Let $E_n(x)$ be Euler polynomial, $\nu_2(n)$ be $2-$adic order of $n,$ $\{g(n)\}$ be the characteristic sequence for $\{2^{n}-1\}_{n\geq1}.$ Recently Peter Luschny asked (cf. \cite{5}, sequence A290646): is for $n\geq1$ A290646=A135517? According to A135517, A091090 and a formula there, this question is equivalent to the following one: is, for $n\geq1,$ the denominator of $E_n(x) - E_n(1)$ equal to $2^{\nu_2(n+1)-g(n)}$? In this note we answer in the affirmative on this question.