I want to define a sequence for the Proth numbers (these are numbers of the form $h\cdot 2^k+1$ for a natural $k$ and an odd $h<2^k$). I want it ordered at first by the exponent, then by the coefficient. So I get the sequence $(a_n)_n = (1\cdot 2^1 + 1, 1\cdot 2^2+1, 3\cdot 2^2+1, 1\cdot 2^3+1\dots).$
But I am really uncertain how to note it. I thought about $$ (a_{k,h})_{k\in\mathbb N \\ 1\leq h \leq 2^k-1, h \text{ odd}}. $$
This looks too much, but I have another problem: If I use double indices, I can't say "the $5-$th number". I must say "the (3,3)-th number".
Is there a way that I can write it down and still refer to an element as the $n$-th element?