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Champernowne constant
The Champernowne constant (named after D. G. Champernowne, and also called Mahler’s number, since Kurt Mahler proved that it is transcendental) is formed by concatenating the positive integers (in base 10) after the decimal point.
Decimal expansion of Champernowne constant
The decimal expansion of Champernowne constant is
- C = 0.1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980...
giving the sequence of decimal digits (A033307)
- {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, ...}
Continued fraction for Champernowne constant
The simple continued fraction for Champernowne constant is
C = 0 +
|
giving the sequence of integer part and partial quotients (A030167)
- {0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4575401113910310764836466282429561185996039397104575550006620043930902626592563149379532077471286563138641209375503552094607183089984575801469863148833592141783010987, 6, ...}
Base b Champernowne constant
The base b Champernowne constant is formed by concatenating the positive integers in baseb, b ≥ 2, |
b |
b |
Cb |
b |
Base 10 expansion
Continued fraction (for base 10 expansion, integer part and partial quotients written in base 10)
0.8622401258680545715577902832493945785657647427682990945160712145573067405905164580420384414386181334...10
{0, 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, 7, 2, 4, 1, 2, 1, 2, 1, 1, 4534532, 1, 4, 5, 1, 2, 1, 7, 1, 16, 1, 4, 1, 5, 5, 1, 5, 1, 4, 1, 2, 1, 5, 3, 2, 38, 2, 12, 1, 15, 2, 6, 3, 30, 4682854730443938, 1, 1, 68, 1, ...}
A066716
A066717
0.59895816753843399250017221792943659097820876867610593675478607547965184195280842055407211080527964157...10
{0, 1, 1, 2, 37, 1, 162, 1, 1, 1, 3, 1, 7, 1, 9, 2, 3, 1, 3068518062211324, 2, 1, 2, 6, 13, 1, 2, 1, 3, 1, 10, 1, 21, 1, 1, 4, 3, 577, 1, 1079268324684171943515797470873767312825026176345571319042096689270, ...}
A077771
A077772
0.???...10
{?, ...}
A??????
A??????
0.???...10
{?, ...}
A??????
A??????
0.???...10
{?, ...}
A??????
A??????
0.???...10
{?, ...}
A??????
A??????
0.???...10
{?, ...}
A??????
A??????
0.???...10
{?, ...}
A??????
A??????
0.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475...10
{0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, ...}
A033307
A030167