# Ambiguity in sequence A073175 First occurrence of an n-digit prime as a substring in the concatenation of the natural numbers

Jonathan Post jvospost3 at gmail.com
Wed Aug 27 21:03:31 CEST 2008

Another clarification to connect several existing and several impliedsequences is as follows.
Axxxxx1 = leftmost n digits of A019518 Smarandache-Wellin numbers:a(n) = concatenation of first n primes = first n digits of the decimal0.23571113171923... (A033308) = Copeland-Erdős constant.
n   a(n)1   22   233   2354   23575   235716   2357117   23571118   235711139   23571113110 2357111317and so forth.
Axxxxx2 = leftmost n digits of Champernowne's constant0.12345678910111213... (A033307) cf. A007908
n   a(n)1   12   123   1234   12345   123456   1234567   12345678   123456789   12345678910 123456789111 1234567891012 12345678910113 1234567891011an so forth.
We can now derive the sequences of the first occurrence of Axxxxx1 inAxxxxx2, and vice versa.
For example:
n   a(n)1   1        1=A007908(1) a substring of 11=prime(5)2   127    12 =A007908(2) a substring of 127=prime(31)3   1231  123=A007908(3) a substring of 1231=prime(202)4   12347  1234=A07908(4) a substring of 12347=prime(1475)5   123457  12345 a substring of prime(11602)and so forth.
This approach fills gaps, allows symmetry, adds cross-references,connects with two interesting real constants (links to their pages inMathWorld), and eliminates ambiguities stemming from digit-countingand presence of zeros.