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A078886 Decimal expansion of Sum {n>=0} 1/5^(2^n). 10

%I #19 Nov 12 2020 01:47:06

%S 2,4,1,6,0,2,5,6,0,0,0,6,5,5,3,6,0,0,0,0,0,0,4,2,9,4,9,6,7,2,9,6,0,0,

%T 0,0,0,0,0,0,0,0,0,0,1,8,4,4,6,7,4,4,0,7,3,7,0,9,5,5,1,6,1,6,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,4,0,2,8,2,3,6,6,9,2,0,9,3,8,4

%N Decimal expansion of Sum {n>=0} 1/5^(2^n).

%C Decimal expansion has increasingly large gaps of zeros, the digits delimited by these zeros are equal to 2^(2^m) as m=0,1,2,3,... The continued fraction expansion (A122165) and consists entirely of 3's, 5's and 7's, after an initial partial quotient of 4. - _Paul D. Hanna_, Aug 22 2006

%H Aubrey J. Kempner, <a href="https://doi.org/10.1090/S0002-9947-1916-1501054-4">On Transcendental Numbers</a>, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals -Sum_{k>=1} mu(2*k)/(5^k - 1), where mu is the Möbius function (A008683). - _Amiram Eldar_, Jul 12 2020

%e 0.241602560006553600000...

%e From _Paul D. Hanna_, Aug 22 2006: (Start)

%e Decimal expansion consists of large gaps of zeros between strings of digits that form powers of 2; this can be seen by grouping the digits as follows:

%e x = .2 4 16 0 256 000 65536 000000 4294967296 000000000000 ...= 0.24160256000655360000004294...

%e and then recognizing the substrings as powers of 2:

%e 2 = 2^(2^0), 4 = 2^(2^1), 16 = 2^(2^2), 65536 = 2^(2^4), 4294967296 = 2^(2^5), 18446744073709551616 = 2^(2^6), ... (End)

%t RealDigits[ N[ Sum[1/5^(2^n), {n, 0, Infinity}], 110]][[1]]

%o (PARI) {a(n)=local(x=sum(k=0,ceil(3+log(n+1)),1/5^(2^k)));(floor(10^n*x))%10} \\ _Paul D. Hanna_, Aug 22 2006

%Y Cf. A122165 (continued fraction), A176594.

%Y Similar sums: A007404, A078885, A078585, A078887, A078888, A078889, A078890, A036987.

%K cons,nonn

%O 0,1

%A _Robert G. Wilson v_, Dec 11 2002

%E Edited by _R. J. Mathar_, Aug 02 2008

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