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 A077473 Greedy powers of (5/9): sum_{n=1..inf} (5/9)^a(n) = 1. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS FORMULA 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 EXAMPLE a(3)=4 since (5/9) +(5/9)^2 +(5/9)^4 < 1 and (5/9) +(5/9)^2 +(5/9)^3 > 1. MATHEMATICA 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] CROSSREFS Cf. A077468, A077469, A077470, A077471, A077472, A077474, A077475. Sequence in context: A174058 A186382 A331417 * A200738 A129896 A134421 Adjacent sequences:  A077470 A077471 A077472 * A077474 A077475 A077476 KEYWORD easy,nonn AUTHOR Paul D. Hanna, Nov 06 2002 EXTENSIONS Edited and extended by Robert G. Wilson v, Nov 08 2002. Also extended by Benoit Cloitre, Nov 06 2002 STATUS approved

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Last modified September 25 19:18 EDT 2021. Contains 347659 sequences. (Running on oeis4.)