A269707 - OEIS

%I #61 Dec 19 2022 04:27:12

%S 3,3,0,0,3,3,0,0,0,0,0,0,0,0,0,0,3,3,0,0,3,3,0,0,0,0,0,0,0,0,0,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,0,0,

%U 3,3,0,0,0,0,0,0,0,0,0,0,3,3,0,0,3,3,0

%N Decimal expansion of x = 3*Sum_{n in E} 1/10^n where E is the set of numbers whose base-4 representation consists of only 0's and 1's.

%C E = {0, 1, 4, 5, 16, 17, 20, 21, 64, ...} (A000695).

%C Among the real numbers it is exceptional for the decimal expansion of a real number to determine the decimal expansion of its reciprocal. The purpose of this sequence is to show an example of such a number.

%C x is irrational. Proof: For all n >= 1, the numbers 3*4^n, 3*4^n + 1, 3*4^n + 2, ..., 3*4^n + 4^(n - 1) each contain at least one base-4 digit different from 0 or 1. So, the decimal expansion of x contains sequences of consecutive zeros with an arbitrary length. Moreover, the decimal expansion also contains an infinite number of digits 3, which implies that x is not periodic, so irrational.

%C We obtain the following property: 1/x = 3*Sum_{n in 2*E} 1/10^(n + 1) where 2*E = {0, 2, 8, 10, 32, 34, 40, 42, ...} (A062880).

%D Daniel Duverney, Number Theory, World Scientific, 2010, 2.10 A striking number, pp. 19-20.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e x = 3.3003300000000003300330000000000000000000000000000...

%e 1/x = 0.303000003030000000000000000000003030000030300000...

%p Digits:=200:nn:=5000:s:=0:

%p for n from 0 to nn do:

%p   x:=convert(n,base,4):n0:=nops(x):

%p   it:=0:ii:=0:

%p     for k from 1 to n0 while(ii=0) do:

%p      if x[k]=0 or x[k]=1

%p       then

%p       it:=it+1:

%p      else

%p     fi:

%p od:

%p if it=n0 then

%p s:= s+evalf(1/10^n):

%p else ii:=1:fi:

%p od:

%p print(3*s):

%p print(1/(3*s)):

%t a[n_] := 3 * Boole[Max @ IntegerDigits[n-1, 4] <= 1]; Array[a, 100] (* _Amiram Eldar_, Aug 06 2021 *)

%Y Cf. A000695, A062880, A151666.

%K nonn,base,cons

%O 1,1

%A _Michel Lagneau_, Mar 10 2016

%E Edited by _Rick L. Shepherd_, May 31 2016