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A077470 Greedy powers of (3/5): Sum_{n>=1} (3/5)^a(n) = 1.

%I A077470 #12 May 22 2025 19:49:14
%S A077470 1,2,7,9,13,15,18,20,22,27,31,37,39,40,49,55,57,66,68,70,71,77,79,81,
%T A077470 82,87,94,98,104,106,107,114,117,120,121,129,133,136,138,141,150,151,
%U A077470 157,158,163,166,169,173,181,184,192,198,199,205,207,209,213,218,224
%N A077470 Greedy powers of (3/5): Sum_{n>=1} (3/5)^a(n) = 1.
%C A077470 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 A077470 A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.5151141759..., where x=3/5 and m=floor(log(1-x)/log(x))=1. - _Paul D. Hanna_, Nov 16 2002
%F A077470 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=3/5 and frac(y) = y - floor(y).
%F A077470 a(n) seems to be asymptotic to c*n with c around 3.7... - _Benoit Cloitre_
%e A077470 a(3)=7 since (3/5) +(3/5)^2 +(3/5)^7 < 1 and (3/5) +(3/5)^2 +(3/5)^6 > 1.
%t A077470 s = 0; a = {}; Do[ If[s + (3/5)^n < 1, s = s + (3/5)^n; a = Append[a, n]], {n, 1, 226}]; a
%t A077470 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[3/5], 20]
%Y A077470 Cf. A077468, A077469, A077471, A077472, A077473, A077474, A077475.
%K A077470 easy,nonn
%O A077470 1,2
%A A077470 _Paul D. Hanna_, Nov 06 2002
%E A077470 Edited and extended by _Robert G. Wilson v_, Nov 08 2002. Also extended by _Benoit Cloitre_, Nov 06 2002