A145572 Numerators of partial sums for Liouville's constant, read as base 2 (binary) numbers.

1, 3, 49, 12845057, 1017690263500988729456314874071089153, 4222921592695952872362526736376161058920018764920519780147745963811744865992371113095993596088044297100172572224585271942341064532181870606866447799704872724575357044373908131956500952542608981420222196042850818326529

Offset: 1

Author

Wolfdieter Lang Mar 06 2009

Keywords

• nonn
• base
• easy

Comments

a(n) is A145571(n) (a decimal number with digits only from {0,1}) read as base 2 number converted back into decimal notation.

The sequence of digit lengths is 1,1,2,8,37,217,1518,... (see A317873).

This sequence gives the numerators of the partial sums for the constant A092874 (called there "binary" Liouville number). See the B(n) formula below. - Wolfdieter Lang, Apr 10 2024

Examples

			a(3)=49, because A145571(3)=110001, and the binary number 110001 translates to 2^5+2^4+2^0=32+16+1 = 49.
		

Crossrefs

Cf. A001563, A092874, A145571 (numerators of approximations for Liouville's number).

Cf. A317873.

Programs

Mathematica

a[n_] := FromDigits[RealDigits[Sum[1/10^k!, {k, n}], 10, n!][[1]], 2]; Array[a, 6] (* Robert G. Wilson v, Aug 08 2018 *)
Block[{k = 0}, NestList[#*2^(++k*k!) + 1 &, 1, 5]] (* Paolo Xausa, Jun 27 2024 *)

Formula

a(n) = A145571(n) interpreted as number in binary notation, then converted to decimal notation.

From Wolfdieter Lang, Apr 10 2024: (Start)

a(n) = Sum_{j=0..n} 2^(n! - j!) = 2^(n!)*B(n) = numerator(B(n)), where B(n) := Sum_{j=1..n} 1/2^(j!), for n >= 1 (Proof from the positions of 1 in A145571).

a(1) = 1, and a(n) = a(n-1)*2^z(n) + 1, where z(n) = n! - (n-1)! = A001563(n-1), for n >= 2.

(End)