Sequences of odd numbers, even numbers and integer squares: gaps in frequency distributions of unit's digits of minor totals

TL;DR

This paper investigates the distribution of units digits in minor totals of sequences like odd, even, and square numbers, revealing digit gaps influenced by numbering systems, especially consistent gaps in odd numbers across bases 3 to 10.

Contribution

It introduces a method using difference tables to predict which digits appear as units in minor totals of key integer sequences, highlighting digit gaps across various numbering systems.

Findings

Gaps in units digit distributions depend on numbering systems.

Odd number sequences show consistent digit gaps across bases 3 to 10.

Difference tables effectively predict units digit appearances.

Abstract

In the paper, I consider appearance of unit's digits in minor totals of a few integer sequences. The sequences include the sequence of even integers, sequence of odd integers and Faulhaber polynomial at $p = 2$. Application of difference tables allows predicting of which digits can appear as unit's digits in minor totals of the sequences. Absence of some digits ("gaps" in frequency distributions) depends often on numbering system applied. However, in case of odd numbers' integers the gaps are found under all numbering systems with bases from 3 to 10.