Copeland and Erdős and a constant

You know the primes? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, … concatenate all the prime numbers and add a 0. before the number and you get 0.23571113171923… or the Copeland-Erdős constant. It is a normal number, which is rare according to Vi Hart.

The constant is given by:

\displaystyle \sum_{n=1}^\infty p_n 10^{-\left(n + \sum_{k=1}^n \lfloor \log_{10}{p_k} \rfloor \right)}

The OEIS sequence A033308 is a sequence of the digits of the Copeland-Erdős constant.

The “extended” version of the Copeland-Erdős constant is Champernowne’s constant, which is every natural number stringed together. So 0.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128…

It’s another normal number. Maybe I should have named this “The Normality of Numbers”. I’ll be sure to do a “part 2”, or, as I stated, “ThE nOrMaLiTy Of NuMbErS”.

For sure.

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