cp's OEIS Frontend

A092874 Decimal expansion of the "binary" Liouville number.

%I A092874 #49 Dec 07 2024 13:15:44
%S A092874 7,6,5,6,2,5,0,5,9,6,0,4,6,4,4,7,7,5,3,9,0,6,2,5,0,0,0,0,0,0,0,0,0,0,
%T A092874 0,0,7,5,2,3,1,6,3,8,4,5,2,6,2,6,4,0,0,5,0,9,9,9,9,1,3,8,3,8,2,2,2,3,
%U A092874 7,2,3,3,8,0,3,9,4,5,9,5,6,3,3,4,1,3,6
%N A092874 Decimal expansion of the "binary" Liouville number.
%C A092874 The famous Liouville number is defined so that its n-th fractional decimal digit is 1 if and only if there exists k, such that k! = n.
%C A092874 The binary Liouville number is defined similarly, but as a binary number: its n-th fractional binary digit is 1 if and only if there exists k, such that k! = n.
%C A092874 According to the definitions introduced in A092855 and A051006, this number is "the binary mapping" of the sequence of factorials (A000142).
%C A092874 For the numerators of the partial sums of B(n) := Sum_{j=1..n} 1/j^(n!) see A145572. - _Wolfdieter Lang_, Apr 10 2024
%D A092874 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.22, p. 172.
%H A092874 G. C. Greubel, <a href="/A092874/b092874.txt">Table of n, a(n) for n = 0..5000</a>
%H A092874 Ferenc Adorján, <a href="https://web.archive.org/web/20221211131908/http://web.axelero.hu/fadorjan/aronsf.pdf">Binary mapping of monotonic sequences and the Aronson function</a>.
%H A092874 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 A092874 Fedoua Sghiouer, Kacem Belhroukia, and Ali Kacha, <a href="https://arxiv.org/abs/2301.06495">Transcendence of some infinite series</a>, arXiv preprint (2023). arXiv:2301.06495 [math.NT].
%H A092874 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A092874 0.7656250596046447753906250000... = 1/2^1 + 1/2^2 + 1/2^6 + 1/2^24 + 1/2^120 + ...
%t A092874 RealDigits[Sum[1/2^(n!), {n, Infinity}], 10, 105][[1]] (* _Alonso del Arte_, Dec 03 2012 *)
%o A092874 (PARI) { mt(v)= /*Returns the binary mapping of v monotonic sequence as a real in (0,1)*/
%o A092874 local(a=0.0,p=1,l);l=matsize(v)[2];
%o A092874 for(i=1,l,a+=2^(-v[i])); return(a)}
%o A092874 (PARI) suminf(n=2,2^-gamma(n)) \\ _Charles R Greathouse IV_, Jun 14 2020
%Y A092874 Cf. A012245, A092855, A051006, A145572.
%K A092874 easy,nonn,cons
%O A092874 0,1
%A A092874 Ferenc Adorjan (fadorjan(AT)freemail.hu)
%E A092874 Offset corrected by _Franklin T. Adams-Watters_, Dec 14 2017