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Unique constant x that satisfies: x = Sum_{n>=1} 1/floor(x^n).
2

%I #18 Jan 17 2018 17:49:53

%S 1,7,7,6,7,9,1,4,2,5,4,8,7,6,5,8,4,2,0,9,9,7,2,9,5,1,2,5,9,3,4,3,7,5,

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

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

%N Unique constant x that satisfies: x = Sum_{n>=1} 1/floor(x^n).

%H Paul D. Hanna, <a href="/A195202/b195202.txt">Table of n, a(n) for n = 1..1971</a>

%F x = Sum_{n>=1} 1/A066173(n) where floor(x^n) = A066173(n).

%e Constant x = 1.7767914254 8765842099 7295125934 3751657100 4017014991 1002131974 4535225732 9321570657 9706460392 2109445017 4890160620 5702665489 ...

%e The integer floor of the powers of the constant x begin:

%e 1=[x], 3=[x^2], 5=[x^3], 9=[x^4], 17=[x^5], 31=[x^6], 55=[x^7], 99=[x^8], 176=[x^9], 313=[x^10], ..., A066173(n) = [x^n], ...

%e where

%e x = 1/1 + 1/3 + 1/5 + 1/9 + 1/17 + 1/31 + 1/55 + 1/99 + 1/176 + 1/313 + 1/557 + 1/990 + 1/1759 + 1/3125 + 1/5553 + 1/9866 +...+ 1/[x^n] +...

%t digits = 100; Clear[s]; s[m_] := s[m] = x /. FindRoot[x == Sum[1/Floor[x^n], {n, 1, m}], {x, 2, 1, 2}, WorkingPrecision -> digits]; s[0] = 0; dm = 100; s[m = dm]; While[ RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m - dm], 10, digits][[1]], m = m + dm; Print[m, " terms"]]; RealDigits[s[m], 10, digits] // First (* _Jean-François Alcover_, Jun 25 2013 *)

%Y Cf. A066173.

%K nonn,cons

%O 1,2

%A _Paul D. Hanna_, Sep 12 2011