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A226160
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Least positive integer k such that 1 + 1/2 + ... + 1/k > n/tau, where tau = golden ratio = (1+sqrt(5))/2.
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1
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1, 2, 4, 7, 12, 23, 42, 79, 146, 271, 503, 934, 1732, 3214, 5963, 11063, 20524, 38078, 70646, 131067, 243166, 451140, 836989, 1552846, 2880960, 5344978, 9916415, 18397696, 34132822, 63325839
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n+1)/a(n) converges to 1.8552...
Conjecture confirmed: using series expansion of HarmonicNumber(k) one gets a(n+1)/a(n) -> exp(1/tau) = 1.855276958... [Jean-François Alcover, Jun 04 2013]
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LINKS
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EXAMPLE
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a(4) = 7 because 1 + 1/2 + ... + 1/6 < 4*tau < 1 + 1/2 + ... + 1/7.
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MATHEMATICA
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nn = 24; g = 1/GoldenRatio; f[n_] := 1/n; a[1] = 1; Do[s = 0; a[n] = NestWhile[# + 1 &, 1, ! (s += f[#]) > n*g &], {n, 1, nn}]; Map[a, Range[nn]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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