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Search: id:A077470
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| A077470 |
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Greedy powers of (3/5): sum_{n=1..inf} (3/5)^a(n) = 1. |
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8
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| 1, 2, 7, 9, 13, 15, 18, 20, 22, 27, 31, 37, 39, 40, 49, 55, 57, 66, 68, 70, 71, 77, 79, 81, 82, 87, 94, 98, 104, 106, 107, 114, 117, 120, 121, 129, 133, 136, 138, 141, 150, 151, 157, 158, 163, 166, 169, 173, 181, 184, 192, 198, 199, 205, 207, 209, 213, 218, 224
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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) = 3.5151141759..., where x=3/5 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2002
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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=(3/5) and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 3.7... - Benoit Cloitre (benoit7848c(AT)orange.fr)
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EXAMPLE
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a(3)=7 since (3/5) +(3/5)^2 +(3/5)^7 < 1 and (3/5) +(3/5)^2 +(3/5)^6 > 1.
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MATHEMATICA
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s = 0; a = {}; Do[ If[s + (3/5)^n < 1, s = s + (3/5)^n; a = Append[a, n]], {n, 1, 226}]; 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[3/5], 20]
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CROSSREFS
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Cf. A077468, A077469, A077471, A077472, A077473, A077474, A077475.
Adjacent sequences: A077467 A077468 A077469 this_sequence A077471 A077472 A077473
Sequence in context: A049638 A059649 A056904 this_sequence A047352 A074244 A102994
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KEYWORD
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easy,nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 06 2002
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EXTENSIONS
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Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 08 2002. Also extended by Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 06 2002
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