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A077468 Greedy powers of (2/3): Sum_{n>=1} (2/3)^a(n) = 1.

%I A077468 #25 May 22 2025 19:06:52
%S A077468 1,3,9,12,15,17,27,34,39,46,49,52,54,66,70,73,81,84,90,95,102,106,110,
%T A077468 116,119,124,132,140,143,149,153,158,161,165,171,177,180,183,186,189,
%U A077468 194,198,209,215,221,224,226,233,235,241,244,248,251,255,259,262,272
%N A077468 Greedy powers of (2/3): Sum_{n>=1} (2/3)^a(n) = 1.
%C A077468 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 A077468 A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n >= m} log(1 + x^n)/log(x) = 4.9298413943..., where x=2/3 and m=floor(log(1-x)/log(x))=2. - _Paul D. Hanna_, Nov 16 2002
%H A077468 Michael Somos, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/NEWS/exp-sqrt">Non-integer Radix Expansions and Modular Functions</a> (1993)
%F A077468 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= 2/3 and frac(y) = y - floor(y).
%F A077468 It appears that, for n>1, a(n) = A073536(n-1) - _Benoit Cloitre_, Jun 04 2004
%e A077468 a(3)=9 since (2/3) +(2/3)^3 +(2/3)^9 < 1 and (2/3) +(2/3)^3 +(2/3)^8 > 1; since the power 8 makes the sum > 1, then 9 is the 3rd greedy power of (2/3).
%t A077468 s = 0; a = {}; Do[ If[s + (2/3)^n < 1, s = s + (2/3)^n; a = Append[a, n]], {n, 1, 278}]; a
%t A077468 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[2/3], 20]
%Y A077468 Cf. A077469, A077470, A077471, A077472, A077473, A077474, A077475.
%K A077468 easy,nonn
%O A077468 1,2
%A A077468 _Paul D. Hanna_, Nov 06 2002
%E A077468 Extended by _John W. Layman_, _Robert G. Wilson v_ and _Benoit Cloitre_, Nov 07 2002