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A077473
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Greedy powers of (5/9): sum_{n=1..inf} (5/9)^a(n) = 1.
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7
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1, 2, 4, 6, 8, 11, 13, 18, 21, 24, 27, 28, 30, 32, 35, 37, 40, 43, 45, 50, 51, 59, 62, 64, 73, 76, 79, 82, 83, 86, 87, 93, 96, 99, 100, 103, 106, 108, 110, 112, 113, 117, 118, 121, 123, 126, 127, 131, 137, 139, 140, 143, 145, 146, 148, 154, 155, 157, 163, 165, 166
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OFFSET
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1,2
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COMMENTS
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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.
A heuristic argument suggests that the limit of a(n)/n is m - sum_{n=m..inf} log(1 + x^n)/log(x) = 2.8326013771..., where x=5/9 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002
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LINKS
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FORMULA
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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=(5/9) and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 2.8... - Benoit Cloitre
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EXAMPLE
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a(3)=4 since (5/9) +(5/9)^2 +(5/9)^4 < 1 and (5/9) +(5/9)^2 +(5/9)^3 > 1.
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MATHEMATICA
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s = 0; a = {}; Do[ If[s + (5/9)^n < 1, s = s + (5/9)^n; a = Append[a, n]], {n, 1, 173}]; a
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[5/9], 20]
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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