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

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

• easy
• nonn
• cons

Comments

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

Examples

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

References

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

Links

• 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.

Crossrefs

Cf. A012245, A092855, A051006, A145572.

Programs

Mathematica

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

PARI

{ 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)}

PARI

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

Extensions

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