# Questions regarding early natural numbers.

Consider the real number $0.123456789101112\dots$, where you concatenate the digits of the natural numbers.

Certain natural numbers are "early", meaning, they appear earlier as a substring of digits than they are supposed to.

For example: $12$, $23$ and $101$ are early natural numbers.

I have a few questions:

1. Is the set of early numbers infinite? What about the set of non-early numbers?

2. Does the set of early numbers have an asymptotic density? If so, is there a closed form expression for that real number?

3. Same as previous question, but regarding the set of non-early numbers.

• The set of early numbers is definitely infinite, since $12,123,1234,12345, \ldots$ are early. – Stefan Mesken Jun 6 '14 at 6:50

## 3 Answers

These numbers are tabulated at the Online Encyclopedia of Integer Sequences. There's a reference there to work of Golomb, which you can find on page 30 of this link. In particular, Golomb explains why these numbers have density 1.

The set of early numbers is infinite, as it contains at least the following set: $$\{12,123,1234,12345,123456,1234567,\dots\}$$

As for the set of non-early numbers, I think you can show that the number $1111\dots 1$ is not early for any number of ones it contains, meaning the set of non-early numbers is also infinite.

Questions 2 and 3 are much tougher and I cannot see a simple answer to them (still, I believe I answered at least some of your questions).

The set of early numbers will definitely go infinite because you can group certain number of digits to achieve a early number. There are also non-early numbers which will go to infinite. There would be always certain numbers (more than early ones) who can't be achieved by the combination of any numbers.