Recursive Equation Indexing

Question

I'm trying to write a recursive equation/formula with all natural numbers as input but I need to exclude every number ending in a $4$ or $9$ ($n= 5i-1$, $i \in \Bbb N)$ and exclude all numbers $n= 13i-1$, $i\in \Bbb N$.

This would leave me with $n=1,2,3,5,6,7,8,10,11,13,15,16,17,18,20\ldots $ to plug into my recursive formula.

Is it possible to write my index with the excluded numbers removed? I'm not sure how to subtract the exclusions from the set of all numbers and write it as a new index. Any help or pointers would be greatly appreciated.

EDIT: Basically, my question is how to rewrite an index. I know that if my index is all natural numbers but I want to exclude the odd numbers ($n= 2m+1, m \in \mathbb{N}$), I can write $l = 2n$ and have an index $l \in \mathbb{N}$. If I want to exclude ($5n-1$) and ($13n-1$), how could I rewrite $n$ so that the index runs through all natural numbers?

Answer 1

I'd go about this by defining the two sets you need to exclude, name them, say, A and B, and end up saying your stuff goes for all $n$ in $\mathbb{N}$ except in both A and B - be it an equation :

$\forall n \in \mathbb{N}\ \backslash\ (A\cup B)\ \ \ function(n) = result(n)$

or a sequence :

$(u_n)_{n\ \in\ \mathbb{N}\ \backslash\ (A\ \cup\ B)}$

or anything else, for all I know. This actually works for any sets $A$, $B$ and indeed any further set that you want to exclude (by further uniting it with the previous ones), and you're free to give any definition of them that will be understood as long as your thing runs.

In your particular case though, there may be a naming convention for the sets you want to exclude - take a peak at congruence classes on $\mathbb{N}$. Granted that the notation is widely accepted, $\overline{a_n}$ describes all integers that a congruent to $a$ modulo $n$. That may spare you the cost of a sentence in English defining A and B before using them. You could write immedialy that you're working on $\mathbb{N}\ \backslash\ (\overline{4_5}\cup\overline{12_{13}})$

Overall though in both cases the notation might appear too heavy. You can always define $C = \mathbb{N}\ \backslash\ (A\cup B)$ and only refer to $C$ afterwards ... But I'm afraid it won't get any better than building up your own working sets.

Hope I helped !

Edit : there's also always the straight-up, upfront, hardcore, sado-masochistic version where you directly say you work on $\mathbb{N}\ \backslash\ (\{n\in \mathbb{N}\ /\ \exists i\in\mathbb{N}\ n=5i-1\}\ \cup\ \{n\in \mathbb{N}\ /\ \exists i\in\mathbb{N}\ n=13i-1\})$. I find it painful, because solely typing it was already quite the pain, but writing it over and over whenever you refer to it will only be that close to hell. Best avoid it, I say.

Edit 2 : I don't know what you know, so with no offense meant I'll specify that for $A$ and $B$ sets, and $P(x)$ a mathematical proposition on mathematical item $x$ :

$A\ \backslash\ B$ is the set of all elements that are in $A$, but not in $B$,

$A\ \cup\ B$ is the set of all elements that are in $A$, or in $B$, or in $A$ and $B$.

$\{n \in A\ /\ P(n) \}$ is the set of elements taken from $A$ that verify property $P$.