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A225569 Decimal expansion of Sum_{n>=0} 1/10^(3^n), a transcendental number.

%I A225569 #34 Nov 05 2023 11:07:24
%S A225569 1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
%T A225569 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A225569 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A225569 Decimal expansion of Sum_{n>=0} 1/10^(3^n), a transcendental number.
%C A225569 According to the Thue-Siegel-Roth theorem, this number is transcendental.
%C A225569 As a sequence, characteristic sequence for powers of 3. - _Franklin T. Adams-Watters_, Aug 07 2013
%C A225569 Actually, characteristic function for 3^k - 1 (A024023), with the current starting offset 0. - _Antti Karttunen_, Nov 19 2017
%D A225569 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 171.
%H A225569 Antti Karttunen, <a href="/A225569/b225569.txt">Table of n, a(n) for n = 0..65537</a>
%H A225569 Wikipedia, <a href="http://en.wikipedia.org/wiki/Thue%E2%80%93Siegel%E2%80%93Roth_theorem">Thue-Siegel-Roth theorem</a>.
%H A225569 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%H A225569 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A225569 From _Antti Karttunen_, Nov 19 2017: (Start)
%F A225569 a(n) = A063524(A053735(1+n)).
%F A225569 a(n) = abs(A154271(1+n)). (End)
%F A225569 From _Amiram Eldar_, Nov 02 2023: (Start)
%F A225569 With offset 1:
%F A225569 Completely multiplicative with a(3^e) = 1, and a(p^e) = 0 for p != 3.
%F A225569 Dirichlet g.f.: 1/(1-3^(-s)). (End)
%e A225569 0.101000001000000000000000001000000000000000000000000000000000000000000000000000001...
%t A225569 (* n = 4 is sufficient to get 100 digits *) Sum[1/10^(3^n), {n, 0, 4}] // RealDigits[#, 10, 100]& // First
%o A225569 (PARI) a(n) = if(n+1 == 3^valuation(n+1, 3), 1, 0); \\ _Amiram Eldar_, Nov 02 2023
%Y A225569 Cf. A000244, A053735, A024023, A036987, A063524, A154271.
%K A225569 nonn,easy,cons
%O A225569 0,1
%A A225569 _Jean-François Alcover_, Jul 29 2013