cp's OEIS Frontend

A012245 Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant.

%I A012245 #94 Feb 16 2025 08:32:32
%S A012245 1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
%T A012245 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 A012245 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A012245 Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant.
%C A012245 Read as decimal fraction 1100010... in any base > 1 (arbitrary decimal point) Liouville's numbers are transcendental; read as a continued fraction it is also transcendental [G. H. Hardy and E. M. Wright, Th. 192].
%D A012245 Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 89.
%D A012245 John H. Conway and Richard K. Guy, The Book of Numbers, pp. 239-241 (1996).
%D A012245 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 162.
%D A012245 T. W. Koerner, Fourier Analysis, Camb. Univ. Press 1988, p. 177.
%D A012245 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 58.
%D A012245 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.
%H A012245 Harry J. Smith, <a href="/A012245/b012245.txt">Table of n, a(n) for n = 1..20000</a>
%H A012245 J. Liouville, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k2977n/f883.image">Communication</a>, C. R. Acad. Sci. Paris 18, 883-885 and 993-995, 1844. [Pages 993-995 do not seem right]
%H A012245 J. Liouville, <a href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1851_1_16_A5_0.pdf">Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques</a>, Journal de Mathématiques Pures et Appliquées 16, pp. 133-142, 1851.
%H A012245 Diego Marques and Carlos Gustavo Moreira, <a href="http://dx.doi.org/10.3792/pjaa.92.39">On variations of the Liouville constant which are also Liouville numbers</a>, Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 3 (2016), 39-40.
%H A012245 Michael Penn, <a href="https://www.youtube.com/watch?v=1sqywCt9tEs">One of the first transcendental numbers -- Liouville's Constant</a>, YouTube video, 2022.
%H A012245 Burkard Polster, <a href="https://www.youtube.com/watch?v=c9nUAXUSuII">Liouville's number, the easiest transcendental and its clones</a>, Mathologer video (2017).
%H A012245 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LiouvillesConstant.html">Liouville's Constant</a>.
%H A012245 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>.
%H A012245 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%H A012245 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>.
%H A012245 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A012245 G.f.: Sum_{i>=1} x^Product_{j=1..i} j. - _Jon Perry_, Mar 31 2004
%F A012245 a(A000142(n)) = 1; a(A063992(n)) = 0. - _Reinhard Zumkeller_, Oct 11 2008
%e A012245 a(25) = a(26) = ... = a(119) = 0 because 4! = 24 and 5! = 120.
%e A012245 0.110001000000000000000001000000000000000000000000000000000000000000000....
%t A012245 With[{nn=5},ReplacePart[Table[0,{nn!}],Table[{n!},{n,nn}]->1]] (* _Harvey P. Dale_, Jul 22 2012 *)
%t A012245 RealDigits[ Sum[1/10^n!, {n, 5}], 10, 105][[1]] (* _Robert G. Wilson v_, Aug 03 2018 *)
%t A012245 CoefficientList[1/x Sum[x^k!, {k, 1, 5}], x] (* _Jean-François Alcover_, Nov 02 2018 *)
%o A012245 (PARI) default(realprecision, 20080); x=10*suminf(n=1, 1.0/10^n!) + 1/10^20040; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b012245.txt", n, " ", d)); \\ _Harry J. Smith_, May 15 2009
%o A012245 (Python)
%o A012245 from itertools import count
%o A012245 def A012245(n):
%o A012245     c = 1
%o A012245     for i in count(1):
%o A012245         if (c:=c*i) >= n:
%o A012245             return int(c==n) # _Chai Wah Wu_, Jan 11 2023
%Y A012245 Cf. A000142, A058304 (continued fraction).
%K A012245 nonn,nice,cons
%O A012245 1,1
%A A012245 _N. J. A. Sloane_