A145572 -id:A145572 - cp's OEIS frontend

A092874 Decimal expansion of the "binary" Liouville number.

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, 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, 7, 2, 3, 3, 8, 0, 3, 9, 4, 5, 9, 5, 6, 3, 3, 4, 1, 3, 6

Offset: 0

Ferenc Adorjan (fadorjan(AT)freemail.hu)

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.
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.
According to the definitions introduced in A092855 and A051006, this number is "the binary mapping" of the sequence of factorials (A000142).
For the numerators of the partial sums of B(n) := Sum_{j=1..n} 1/j^(n!) see A145572. - Wolfdieter Lang, Apr 10 2024

			0.7656250596046447753906250000... = 1/2^1 + 1/2^2 + 1/2^6 + 1/2^24 + 1/2^120 + ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.22, p. 172.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Ferenc Adorján, Binary mapping of monotonic sequences and the Aronson function.
Burkard Polster, Liouville's number, the easiest transcendental and its clones, Mathologer video (2017).
Fedoua Sghiouer, Kacem Belhroukia, and Ali Kacha, Transcendence of some infinite series, arXiv preprint (2023). arXiv:2301.06495 [math.NT].
Index entries for transcendental numbers.

Cf. A012245, A092855, A051006, A145572.

RealDigits[Sum[1/2^(n!), {n, Infinity}], 10, 105][[1]] (* Alonso del Arte, Dec 03 2012 *)

{ mt(v)= /*Returns the binary mapping of v monotonic sequence as a real in (0,1)*/
local(a=0.0,p=1,l);l=matsize(v)[2];
for(i=1,l,a+=2^(-v[i])); return(a)}

suminf(n=2,2^-gamma(n)) \\ Charles R Greathouse IV, Jun 14 2020

Offset corrected by Franklin T. Adams-Watters, Dec 14 2017

A317873 Number of digits in 2^(n!).

Original entry on oeis.org

1, 1, 1, 2, 8, 37, 217, 1518, 12138, 109238, 1092378, 12016155, 144193850, 1874520045, 26243280622, 393649209329, 6298387349264, 107072584937472, 1927306528874488, 36618824048615255, 732376480972305082, 15379906100418406713, 338357934209204947674

Offset: 0

Wolfdieter Lang and Robert G. Wilson v, Aug 09 2018

The old definition (which did not match the data) was "Number of digits in the numerators of partial sums for Liouville's constant, read as base-2 (binary) numbers (A145572)."

Kevin Ryde, Table of n, a(n) for n = 0..300

Cf. A000142, A034887, A050923, A055642, A145572.

Digits := 900: # for n <= 300
a := n -> ceil(exp(lnGAMMA(n + 1))*log10(2)):
seq(a(n), n = 0..30);  # Peter Luschny, Apr 18 2024

Mathematica
```
Array[ Floor[#! Log10@2 + 1] &, 22]
```

a(n) = A034887(n!).

Better definition suggested by Martin Renner, Mar 24 2024

a(0)=1 prepended by Alois P. Heinz, Jul 27 2025

A145571 Numerators of partial sums for Liouville's constant.

Original entry on oeis.org

1, 11, 110001, 110001000000000000000001, 110001000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

Offset: 1

Wolfdieter Lang, Mar 06 2009

The denominators are 10^(n!).
In a(n) the 1's appear at positions j!, j=1..n. Therefore Liouville's constant c:=Sum_{k>=1} 1/10^(k!) is the number 0.a(n) with n -> infinity.
Liouville's constant c is transcendental. See, e.g., the proof in the Rosenberger-Fine reference.
The number of digits of a(n) is n! = A000142(n). The number of 0s is 0 for n = 1 and 2, and Sum_{k=3..n} (z(n) - 1), for n >= 3, where z(n) = n! - (n-1)! = A001563(n-1). This number is n! - n, for n >= 1. - Wolfdieter Lang, Apr 09 2024

			a(2)=11 because c(2)=1/10 + 1/100 = 11/100.
a(6) has 1's at positions 1,2,6,24,120,720 (A000142, factorials) and 0's in between.

B. Fine and G. Rosenberger, Number theory: an introduction via the distribution of primes, Birkhäuser, Boston, Basel, Berlin, 2007. Th. 6.3.2.3., p. 286.

Cf. A000142, A001563, A145572 (a(n) read as base 2 representation).

Numerator[Accumulate[1/10^Range[6]!]] (* Paolo Xausa, Jun 25 2024 *)
Block[{k = 0}, NestList[#*10^(++k*k!) + 1 &, 1, 5]] (* Paolo Xausa, Jun 26 2024 *)

a(n) = numerator(c(n)), with c(n):= Sum_{k=1..n} 1/10^(k!).

a(1) = 1, and a(n) = a(n-1)*10^(z(n)) + 1, for n >= 2, where z(n) = A001563(n-1) = n! - (n-1)! = (n-1)!*(n-1). - Wolfdieter Lang, Apr 09 2024 [Corrected by Paolo Xausa, Jun 26 2024 ]

cp's OEIS Frontend

A092874 Decimal expansion of the "binary" Liouville number.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A317873 Number of digits in 2^(n!).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A145571 Numerators of partial sums for Liouville's constant.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula