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Search: id:A080059
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| A080059 |
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Greedy powers of (1/zeta(3)): sum_{n=1..inf} (1/zeta(3))^a(n) = 1, where 1/zeta(3) = .83190737258070746868... |
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2
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| 1, 10, 26, 38, 54, 64, 80, 98, 115, 126, 136, 147, 158, 171, 181, 196, 206, 226, 243, 257, 267, 279, 293, 306, 324, 334, 355, 365, 378, 388, 398, 410, 432, 442, 455, 468, 491, 501, 519, 534, 545, 560, 572, 582, 593, 610, 628, 638, 650, 663, 672, 691, 704, 715
(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) = 14.874449248373..., where x=(1/zeta(3)) and m=floor(log(1-x)/log(x))=9.
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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=(1/zeta(3)) and frac(y) = y - floor(y). See A077468 for mathematica program by Robert G. Wilson v.
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EXAMPLE
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a(3)=26 since (1/zeta(3)) +(1/zeta(3))^10 +(1/zeta(3))^26 < 1 and (1/zeta(3)) +(1/zeta(3))^10 +(1/zeta(3))^k > 1 for 10<k<26.
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CROSSREFS
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Cf. A077468, A080059.
Sequence in context: A005278 A157075 A045039 this_sequence A071348 A055042 A044071
Adjacent sequences: A080056 A080057 A080058 this_sequence A080060 A080061 A080062
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre (benoit7848c(AT)orange.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Jan 23 2003
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