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%I #10 Mar 30 2012 18:39:11
%S 1,5,16,21,29,35,39,52,57,63,68,76,82,88,93,99,106,113,118,127,134,
%T 150,155,160,167,172,182,192,197,209,215,224,229,237,242,246,260,265,
%U 272,278,289,293,310,315,320,330,337,346,353,373,379,384,390,396,405,416
%N Greedy powers of (3/4): sum_{n=1..inf} (3/4)^a(n) = 1.
%C 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 A heuristic argument suggests that the limit of a(n)/n is m - sum_{n=m..inf} log(1 + x^n)/log(x) = 8.0547475948..., where x=3/4 and m=floor(log(1-x)/log(x))=4. - _Paul D. Hanna_, Nov 16 2002
%F a(n)=sum_{k=1..n}floor(g_k) where g_1=1, g_{n+1}=log_x(x^frac(g_n) - x) (n>0) at x=(3/4) and frac(y) = y - floor(y).
%F a(n) seems to be asymptotic to c*n with c around 8.0... - _Benoit Cloitre_
%e a(3)=9 since (3/4) +(3/4)^5 +(3/4)^16 < 1 and (3/4) +(3/4)^5 +(3/4)^15 > 1.
%t s = 0; a = {}; Do[ If[s + (3/4)^n < 1, s = s + (3/4)^n; a = Append[a, n]], {n, 1, 428}]; a
%t 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[3/4], 20]
%Y Cf. A077468, A077470, A077471, A077472, A077473, A077474, A077475.
%K easy,nonn
%O 1,2
%A _Paul D. Hanna_, Nov 06 2002
%E Edited and extended by _Robert G. Wilson v_, Nov 08 2002. Also extended by _Benoit Cloitre_, Nov 06 2002
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