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Search: id:A077471
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| A077471 |
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Greedy powers of (4/7): sum_{n=1..inf} (4/7)^a(n) = 1. |
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8
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| 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; 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) = 3.0486255758..., where x=4/7 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=(4/7) and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 3.3... - Benoit Cloitre (benoit7848c(AT)orange.fr)
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EXAMPLE
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a(3)=5 since (4/7) +(4/7)^2 +(4/7)^5 < 1 and (4/7) +(4/7)^2 +(4/7)^4 > 1.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A077468, A077469, A077470, A077472, A077473, A077474, A077475.
Sequence in context: A109150 A004202 A013647 this_sequence A064572 A032399 A007969
Adjacent sequences: A077468 A077469 A077470 this_sequence A077472 A077473 A077474
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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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