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Normal or not? (Posted on 2012-04-29) Difficulty: 3 of 5

The Champernowne constant, C10, can easily be formed by concatenating the natural numbers to produce the real number: 0.123456789101112... etc., see e.g. here.

But this is a bit wasteful, for as young Hannah Rollman (HR) has pointed out, we can afford to 'write down the numbers 1,2,3,... but omit any number (such as 12 or 23 or 31 ...) which has appeared as a string earlier in the sequence,' see e.g. here.

If we concatenate the HR numbers to produce a real number we obtain: 0.12345678910111314...etc. Call this the 'HR constant'.

It is known that C10 is "10-normal". (actually it was made up specifically so as to meet that criterion) i.e. in its decimal expansion the digit 7 must appear 1/10 of the time, the string 783 must appear 1/1000 of the time, and so on.

Question: is the HR constant "10-normal"?

(Inspired by thinking about 'Loops in Pi'.)

No Solution Yet Submitted by broll    
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I think... | Comment 1 of 3
I think the answer may be no.

Writing out 1 through 99, there are 36 numbers that are omitted.  Each of the digits 1 through 9 is omitted 8 times, but the 0 never is. 

Continuing on from 100 through 199, 1 is omitted 22 times, 2 is omitted 3 times 3 is omitted 3 times, and 4 through 9 once each. 

Continuing up to 999 it looks like 0 will be omitted less often then the others, but I'm not positive and I haven't actually checked.

  Posted by Jer on 2012-05-01 12:52:07
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