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 A077471 Greedy powers of (4/7): sum_{n=1..inf} (4/7)^a(n) = 1. 8
 1, 2, 5, 6, 10, 11, 14, 18, 19, 23, 27, 29, 30, 35, 36, 39, 55, 56, 60, 62, 64, 73, 75, 78, 79, 83, 84, 87, 95, 99, 104, 111, 113, 121, 122, 126, 133, 134, 141, 143, 147, 151, 152, 161, 162, 165, 169, 171, 173, 175, 176, 179, 182, 183, 186, 189, 197, 202, 205, 207 (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) = 3.0486255758..., where x=4/7 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 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=(4/7) and frac(y) = y - floor(y). a(n) seems to be asymptotic to c*n with c around 3.3... - Benoit Cloitre EXAMPLE a(3)=5 since (4/7) +(4/7)^2 +(4/7)^5 < 1 and (4/7) +(4/7)^2 +(4/7)^4 > 1. MAPLE s:= 0: count:= 0: R:= NULL; for n from 1 while count < 100 do   t:= (4/7)^n;   if s+t < 1 then count:= count+1; R:= R, n; s:= s+t fi od: R; # Robert Israel, Jun 01 2018 MATHEMATICA s = 0; a = {}; Do[ If[s + (4/7)^n < 1, s = s + (4/7)^n; a = Append[a, n]], {n, 1, 208}]; 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[4/7], 20] CROSSREFS Cf. A077468, A077469, A077470, A077472, A077473, A077474, A077475. Sequence in context: A187840 A187904 A179543 * A273324 A238096 A064572 Adjacent sequences:  A077468 A077469 A077470 * A077472 A077473 A077474 KEYWORD easy,nonn AUTHOR Paul D. Hanna, Nov 06 2002 EXTENSIONS Extended by Benoit Cloitre, Nov 06 2002 Edited and extended by Robert G. Wilson v, Nov 08 2002 STATUS approved

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Last modified July 27 23:16 EDT 2021. Contains 346316 sequences. (Running on oeis4.)