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A077471 Greedy powers of (4/7): Sum_{n>=1} (4/7)^a(n) = 1.

%I A077471 #18 May 22 2025 20:03:54
%S A077471 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 A077471 79,83,84,87,95,99,104,111,113,121,122,126,133,134,141,143,147,151,
%U A077471 152,161,162,165,169,171,173,175,176,179,182,183,186,189,197,202,205,207
%N A077471 Greedy powers of (4/7): Sum_{n>=1} (4/7)^a(n) = 1.
%C A077471 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 A077471 A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} 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 A077471 Robert Israel, <a href="/A077471/b077471.txt">Table of n, a(n) for n = 1..10000</a>
%F A077471 a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g_n) - x) at x=4/7 and frac(y) = y - floor(y).
%F A077471 a(n) seems to be asymptotic to c*n with c around 3.3... - _Benoit Cloitre_
%e A077471 a(3)=5 since (4/7) +(4/7)^2 +(4/7)^5 < 1 and (4/7) +(4/7)^2 +(4/7)^4 > 1.
%p A077471 s:= 0: count:= 0:
%p A077471 R:= NULL;
%p A077471 for n from 1 while count < 100 do
%p A077471   t:= (4/7)^n;
%p A077471   if s+t < 1 then count:= count+1; R:= R, n; s:= s+t fi
%p A077471 od:
%p A077471 R; # _Robert Israel_, Jun 01 2018
%t A077471 s = 0; a = {}; Do[ If[s + (4/7)^n < 1, s = s + (4/7)^n; a = Append[a, n]], {n, 1, 208}]; a
%t A077471 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 A077471 Cf. A077468, A077469, A077470, A077472, A077473, A077474, A077475.
%K A077471 easy,nonn
%O A077471 1,2
%A A077471 _Paul D. Hanna_, Nov 06 2002
%E A077471 Extended by _Benoit Cloitre_, Nov 06 2002
%E A077471 Edited and extended by _Robert G. Wilson v_, Nov 08 2002