Suppose there are $n$ terms of an arithmetic progression of the form
$$(2i+1)k + x_i, ~~~~ i = 1,\ldots,n, ~~~~ k \geq 0,$$
for varying initial integer terms $x_i \geq 0$.
The problem is to cover $m(n)$—the maximum possible number of consecutive integers starting with $1$ (the number is covered, if it belongs to at least one of these progressions).
For example, is it true that $$m(n) \approx \operatorname*{LCM}\limits_{i=1,\,\ldots\,,\,n}(\{2i+1\}) \text{?}$$
Numerical simulations seem to suggest $m(n) \approx Cn$.