Long Solutions of Sequence A348480 of the On-Line Encyclopedia of Integer Sequences

TL;DR

This paper investigates special binary solutions related to sequence A348480, focusing on long solutions where the binary number contains many zeros, and provides proofs for certain cases.

Contribution

It presents proofs that numbers 7 and 13 have long solutions, expanding understanding of binary solutions in sequence A348480.

Findings

Numbers 11 and 37 have long solutions due to their porosity.

Proofs for the existence of long solutions for 7 and 13 are provided.

Most minimal solutions contain few zeros, unlike long solutions.

Abstract

For numbers $x$ coprime to $10$ there exist infinitely many binary numbers $b$ such that the greatest common divisor of $b$ and rev($b$) = $x$ and the sum of digits of $b = x$ (rev($b$) is the digit reversal of $b$). In most cases, the smallest $b$ that fulfill these two constraints contain just a few zeros. But in some cases like for $x = 7, 11, 13$ and $37$, $b$ must contain more zeros than ones and these $b$ are called long solutions. For $11$ and $37$ it follows directly from the fact that these are porous numbers. For $7$ and $13$, the proofs that they have long solutions are presented in this paper.