Formally, the condition isn't "never use a one-time pad with the same key numbers" it's "make sure the key numbers of each pad are statistically independent of each other".
Not that it matters. The sqrt(n) birthday effect isn't sufficient to overcome the exponentially low chance of a one-time pad collision for pads of any substantial size.
> Formally, the condition isn't "never use a one-time pad with the same key numbers" it's "make sure the key numbers of each pad are statistically independent of each other".
And to expand on this - this is why it is imperative that a source of true randomness is used as a seed. Pseudorandomness cannot (by definition) satisfy statistical independence.
If you don't have a source of true randomness, you've just implemented a stream cipher, not a one-time pad.
Not that it matters. The sqrt(n) birthday effect isn't sufficient to overcome the exponentially low chance of a one-time pad collision for pads of any substantial size.