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A077475 Greedy powers of (8/13): Sum_{n>=1} (8/13)^a(n) = 1.

%I A077475 #18 May 22 2025 22:07:07
%S A077475 1,2,11,14,25,28,30,37,39,41,43,46,48,51,54,57,60,64,66,71,76,78,80,
%T A077475 82,84,90,95,101,103,106,110,113,115,117,127,133,135,140,146,152,157,
%U A077475 160,162,165,167,170,173,179,181,185,189,196,200,203,206,209,212,215,220
%N A077475 Greedy powers of (8/13): Sum_{n>=1} (8/13)^a(n) = 1.
%C A077475 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 A077475 A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.8170308430..., where x=8/13 and m=floor(log(1-x)/log(x))=1. - _Paul D. Hanna_, Nov 16 2002
%C A077475 By the time you reach Sum_{n=1..59} (8/13)^a(n), the difference between that sum and 1 is only 1.6*10^-47.
%H A077475 Robert Israel, <a href="/A077475/b077475.txt">Table of n, a(n) for n = 1..10000</a>
%F A077475 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=8/13 and frac(y) = y - floor(y).
%F A077475 a(n) seems to be asymptotic to c*n with c around 3.7... - _Benoit Cloitre_
%e A077475 a(3)=11 since (8/13) +(8/13)^2 +(8/13)^11 < 1 and (8/13)+(8/13)^2+(8/13)^10 >1.
%p A077475 V:= Vector(100):
%p A077475 V[1]:= 1: T:= 1 - 8/13:
%p A077475 for n from 2 to 100 do
%p A077475 V[n]:= -floor(log[13/8](T));
%p A077475 T:= T - (8/13)^V[n];
%p A077475 od:
%p A077475 convert(V,list); # _Robert Israel_, Aug 11 2020
%t A077475 s = 0; a = {}; Do[ If[s + (8/13)^n < 1, s = s + (8/13)^n; a = Append[a, n]], {n, 1, 250}]; a
%t A077475 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[8/13], 20]
%Y A077475 Cf. A077468, A077469, A077470, A077471, A077472, A077473, A077474.
%K A077475 easy,nonn
%O A077475 1,2
%A A077475 _Paul D. Hanna_, Nov 06 2002
%E A077475 Edited and extended by _Robert G. Wilson v_, Nov 08 2002.
%E A077475 Extended by _Benoit Cloitre_, Nov 06 2002