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A269707 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.

%I A269707 #61 Dec 19 2022 04:27:12
%S A269707 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 A269707 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 A269707 3,3,0,0,0,0,0,0,0,0,0,0,3,3,0,0,3,3,0
%N A269707 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 A269707 E = {0, 1, 4, 5, 16, 17, 20, 21, 64, ...} (A000695).
%C A269707 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 A269707 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 A269707 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 A269707 Daniel Duverney, Number Theory, World Scientific, 2010, 2.10 A striking number, pp. 19-20.
%H A269707 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e A269707 x = 3.3003300000000003300330000000000000000000000000000...
%e A269707 1/x = 0.303000003030000000000000000000003030000030300000...
%p A269707 Digits:=200:nn:=5000:s:=0:
%p A269707 for n from 0 to nn do:
%p A269707   x:=convert(n,base,4):n0:=nops(x):
%p A269707   it:=0:ii:=0:
%p A269707     for k from 1 to n0 while(ii=0) do:
%p A269707      if x[k]=0 or x[k]=1
%p A269707       then
%p A269707       it:=it+1:
%p A269707      else
%p A269707     fi:
%p A269707 od:
%p A269707 if it=n0 then
%p A269707 s:= s+evalf(1/10^n):
%p A269707 else ii:=1:fi:
%p A269707 od:
%p A269707 print(3*s):
%p A269707 print(1/(3*s)):
%t A269707 a[n_] := 3 * Boole[Max @ IntegerDigits[n-1, 4] <= 1]; Array[a, 100] (* _Amiram Eldar_, Aug 06 2021 *)
%Y A269707 Cf. A000695, A062880, A151666.
%K A269707 nonn,base,cons
%O A269707 1,1
%A A269707 _Michel Lagneau_, Mar 10 2016
%E A269707 Edited by _Rick L. Shepherd_, May 31 2016