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This probably isn't the most important question, but I'm working on something where I've found it convenient to define the following two functions (on the integers): $$ I_o(q):= \begin{cases} q & q \text{ is odd } \\ q-1 & q \text{ is even} \end{cases} \qquad \text{ and } \qquad I_e(q):= \begin{cases} q & q \text{ is even } \\ q-1 & q \text{ is odd} \end{cases}. $$

So $I_o(q)$ is just the largest odd number less than or equal to $q$ and $I_e(q)$ is the same, except for even numbers. I feel that I've seen this function before with a notation similar as to that of floor, but I've had little luck googling it. So I was curious if there was a common notation for this function.

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You have

$$I_o(q) = 2 \left\lfloor \frac{q-1}{2} \right\rfloor + 1 = \left\{ \begin{array}{lc} q & q \text{ is odd} \\ q-1 & q \text{ is even} \end{array} \right.$$


$$ I_e(q) = 2 \left\lfloor \frac{q}{2} \right\rfloor. $$

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This gives a formula for the functions, but OP was asking for a common notation (if there is one). – Gerry Myerson Jul 29 '11 at 6:12

I don't know if I'd call this common notation, but Microsoft Office seems to have a function EVEN(x) that rounds $x$ up to the nearest even integer. Of course, Microsoft has a strange definition of "up", as EVEN(-1) is -2, not 0. See this MS Office link.

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