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Binary expansion of Sum_{n>=1} A006218(n)*2^(-n).
4

%I #13 Aug 01 2020 11:36:14

%S 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,

%T 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,

%U 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

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

%C 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

%H David H. Bailey and Richard E. Crandall, <a href="https://projecteuclid.org/euclid.em/1057864662">Random generators and normal numbers</a>, Experimental Mathematics, Vol. 11, No. 4 (2002), pp. 527-546. See p. 540.

%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1948-04.pdf">On Arithmetical Properties of Lambert Series</a>, J. Indian Math. Soc., Vol. 12 (1948), 63-66.

%e 11.00110110101000001011111100111100100001...

%t f[n_, m_] := Sum[Floor[n/k], {k, 1, m}]

%t t = Table[f[n, 100], {n, 1, 4000}] ;

%t N[Sum[t[[n]]/2^n, {n, 1, 4000}], 100]

%t RealDigits[%, 10] (* A211705 *)

%t RealDigits[%%, 2] (* A211706 *)

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

%K nonn,cons,base

%O 2

%A _Clark Kimberling_, Apr 19 2012

%E Offset changed from _Bruno Berselli_, May 14 2012