On Numbers of the Form 5^k 2^m - 1 Generated from Mersenne Numbers

Abstract

We present a numerical experiment in which the neighbour-sum transform T, defined as (T a)_n = a_{n-1} + a_{n+1}, is applied to Mersenne numbers M_n = 2^n - 1, followed by repeated division by powers of 2. After several iterations, a new class of numbers emerges: those of the form 5^k * 2^m - 1, where k and m are integers. We derive a simple inverse formula that recovers the original Mersenne number from any number in this class. The result is verified with extensive tables and Python code. The discovered sequence was checked against the OEIS database and no match was found, suggesting that this class is novel. Additional structural properties are observed: a transient region whose length equals the number of applied transforms, and the consistent appearance of the digit 9 in all regular terms. The results are fully reproducible using the provided code.