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Greedy powers of (4/7): sum_{n=1..inf} (4/7)^a(n) = 1.
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%I #16 Jun 01 2018 04:21:16

%S 1,2,5,6,10,11,14,18,19,23,27,29,30,35,36,39,55,56,60,62,64,73,75,78,

%T 79,83,84,87,95,99,104,111,113,121,122,126,133,134,141,143,147,151,

%U 152,161,162,165,169,171,173,175,176,179,182,183,186,189,197,202,205,207

%N Greedy powers of (4/7): sum_{n=1..inf} (4/7)^a(n) = 1.

%C The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series sum_{k=1..n} x^a(k) to exceed unity.

%C A heuristic argument suggests that the limit of a(n)/n is m - sum_{n=m..inf} log(1 + x^n)/log(x) = 3.0486255758..., where x=4/7 and m=floor(log(1-x)/log(x))=1. - _Paul D. Hanna_, Nov 16 2002

%H Robert Israel, <a href="/A077471/b077471.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} floor(g_k) where g_1=1, g_{n+1}=log_x(x^frac(g_n) - x) (n>0) at x=(4/7) and frac(y) = y - floor(y).

%F a(n) seems to be asymptotic to c*n with c around 3.3... - _Benoit Cloitre_

%e a(3)=5 since (4/7) +(4/7)^2 +(4/7)^5 < 1 and (4/7) +(4/7)^2 +(4/7)^4 > 1.

%p s:= 0: count:= 0:

%p R:= NULL;

%p for n from 1 while count < 100 do

%p t:= (4/7)^n;

%p if s+t < 1 then count:= count+1; R:= R, n; s:= s+t fi

%p od:

%p R; # _Robert Israel_, Jun 01 2018

%t s = 0; a = {}; Do[ If[s + (4/7)^n < 1, s = s + (4/7)^n; a = Append[a, n]], {n, 1, 208}]; a

%t heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[4/7], 20]

%Y Cf. A077468, A077469, A077470, A077472, A077473, A077474, A077475.

%K easy,nonn

%O 1,2

%A _Paul D. Hanna_, Nov 06 2002

%E Extended by _Benoit Cloitre_, Nov 06 2002

%E Edited and extended by _Robert G. Wilson v_, Nov 08 2002