tables > nrr

Near Repdigit Related

This section is inspired by Studio Kamada: Factorization of near-repdigit-related numbers. Numbers of the following forms (in base 10) are selected:

For this database, these types of numbers are extended to bases 2-36, however the number of patterns is proportional to the base cubed, so many larger ones will not make it into here for a long time. For base 10, there are 603 unique forms after removing duplicates and factorable sequences. For base 36, there are over 27000.

By using some algebra with geometric summations, we can find that each such pattern in base \(b\) is a rational number times a polynomial of \(b^n\) with integer coefficients.

Some such near repdigit patterns are not stored either because they are equivalent to another under shift and multiply or because their terms factor into the terms of other near repdigit patterns. In particular, 2 sequences \(f,g\) are considered equivalent if we can find some \(k\in\mathbb{Q}\) and \(s\in\mathbb{Z}\) such that \(f(n)=k\cdot g(n+s)\). A sequence \(f\) is factorable into \(g,h\) if we can find some \(k\in\mathbb{Q}\) and \(s_1,s_2\in\mathbb{Z}\) such that \(f(n)=k\cdot g(n+s_1)\cdot h(n+s_2)\) (possibly the product of even more than 2, but that never occurs for any of the types of sequences selected so far). What this means practically for the goal of factoring near repdigit sequences is that the work to factor some is essentially equivalent to the work of factoring one or more other sequences.

The notation used for such sequences on this website can be explained by an example: \[[4^*1_{10}]_n=\{1,41,441,4441,44441,\ldots\}\] The subscript \(n\) at the end means the \(n\)th term, all sequences are defined for nonnegative integers \(n\). The subscript \(10\) means base 10. The digit sequence is represented by the digits (using \(\text{a}-\text{z}\) for going up to base 36) and a superscript \(*\) means to repeat the digit \(n\) times. This example has \(4\) repeated \(n\) times followed by a single \(1\). There can be multiple digits with a superscript \(*\). Each superscript \(*\) means repeat \(n\) times (same \(n\) for each occurrence). Although the notation is inspired by regular expressions, these are technically not regular expressions if there are multiple superscript \(*\) because each repetition count must be the same.

Links

> Base 2 (category) > Base 3 (category) > Base 4 (category) > Base 5 (category) > Base 6 (category) > Base 7 (category) > Base 8 (category) > Base 9 (category) > Base 10 (category)