Symbol for the set of odd naturals?

Obviously the set of naturals is denoted $\mathbb{N}$, but is there a symbol for the set of odd naturals? Would $2\mathbb{N}+1$ (or $2\mathbb{N}-1$) be a standard notation?

I don't recall seeing too many places that gave a specific notation to the set of even or odd numbers. Your notation of $2\mathbb N+1$ seems quite reasonable.

As with all notational problems, my usual tip is to find something that seems reasonable and simply declare it in the first few lines (or when you need to use it):

Let $\mathbb O$ denote the set of positive odd integers.

How about $\mathbb{N}\setminus2\mathbb{N}$ ?

Why not $\{2n-1,\;n\in\mathbb{N}\}?$ Or are you looking for an actual symbol? I recall seeing $\mathbb{E}$ and $\mathbb{O}$ somewhere - (of course, specify what you mean if you use the $\mathbb{E}$'s $\mathbb{O}$'s $\mathbb{P}$'s etc. in a paper).