A211706 Binary expansion of Sum_{n>=1} A006218(n)*2^(-n).

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0

Offset

2

Comments

With offset 1 this is the binary expansion of the Erdős-Borwein constant (A065442). Erdős (1948) proved that this constant is irrational by showing that its binary expansion has arbitrarily long strings of zeros. - Amiram Eldar, Aug 01 2020

Links

Table of n, a(n) for n=2..100.

David H. Bailey and Richard E. Crandall, Random generators and normal numbers, Experimental Mathematics, Vol. 11, No. 4 (2002), pp. 527-546. See p. 540.

John M. Campbell, On the binary digits of the Erdős-Borwein constant, arXiv:2605.24160 [math.NT], 2016.

Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc., Vol. 12 (1948), 63-66.

Example

11.00110110101000001011111100111100100001...

Mathematica

f[n_, m_] := Sum[Floor[n/k], {k, 1, m}]
t = Table[f[n, 100], {n, 1, 4000}] ;
N[Sum[t[[n]]/2^n, {n, 1, 4000}], 100]
RealDigits[%%, 2]  (* A211706 *)

Crossrefs

Cf. A006218, A065442, A211701, A211705 (decimal representation)

Sequence in context: A265698 A267673 A267845 * A267683 A267869 A068434

Adjacent sequences: A211703 A211704 A211705 * A211707 A211708 A211709

Keyword

nonn,cons,base

Author

Clark Kimberling, Apr 19 2012

Extensions

Offset changed from Bruno Berselli, May 14 2012

Status

approved