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A077472 Greedy powers of (5/8): Sum_{n>=1} (5/8)^a(n) = 1.

%I A077472 #12 May 22 2025 21:29:52
%S A077472 1,3,5,8,10,13,15,23,26,30,33,36,38,46,48,51,53,57,61,64,66,69,72,76,
%T A077472 78,84,88,93,95,104,106,110,115,117,121,126,129,131,136,138,143,148,
%U A077472 150,152,157,160,164,169,172,175,179,181,185,187,191,196,198,201,203
%N A077472 Greedy powers of (5/8): Sum_{n>=1} (5/8)^a(n) = 1.
%C A077472 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 A077472 A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.9944918847..., where x=5/8 and m=floor(log(1-x)/log(x))=2. - _Paul D. Hanna_, Nov 16 2002
%F A077472 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=(5/8) and frac(y) = y - floor(y).
%F A077472 a(n) seems to be asymptotic to c*n with c around 3.4... - _Benoit Cloitre_
%e A077472 a(3)=5 since (5/8) +(5/8)^3 +(5/8)^5 < 1 and (5/8) +(5/8)^3 +(5/8)^4 > 1.
%t A077472 s = 0; a = {}; Do[ If[s + (5/8)^n < 1, s = s + (5/8)^n; a = Append[a, n]], {n, 1, 210}]; a
%t A077472 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[5/8], 20]
%Y A077472 Cf. A077468, A077469, A077470, A077471, A077473, A077474, A077475.
%K A077472 easy,nonn
%O A077472 1,2
%A A077472 _Paul D. Hanna_, Nov 06 2002
%E A077472 Edited and extended by _Robert G. Wilson v_, Nov 08 2002. Also extended by _Benoit Cloitre_, Nov 06 2002