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Search: id:A004796
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| A004796 |
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n-th positive integer k such that if 2 <= j < k then the fractional part of the k-th partial sum of harmonic series is < the fractional part of the j-th partial sum of the harmonic series. |
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+0
2
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| 4, 11, 83, 616, 1674, 4550, 12367, 33617, 91380, 248397, 1835421, 4989191, 13562027, 36865412, 272400600, 740461601
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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n such that H(n) sets a new record for being a tiny bit greater than an integer, where H(n)=sum_{k=1..n} 1/k. For proofs that H(n) is non-integral and almost always a non-terminating decimal see Havil reference.
Assuming that H(n) ~= Ln(n) + gamma + 1/(2n), the next several entries should be 2012783315, 5471312310 and 40427833596; 14872568831 and 109894245429 are not included. - Robert G. Wilson v.
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REFERENCES
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Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 24-25, 2003.
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LINKS
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T. Sillke, The Harmonic Numbers and Series
Eric Weisstein's World of Mathematics, Harmonic Series.
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EXAMPLE
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a(2)=11 because H(11) = 3.0198773...; a(3)=83 because H(83) = 5.0020682...
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MATHEMATICA
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s = 0; a = 1; Do[ s = N[s + 1/n, 50]; If[ FractionalPart[s] < a, a = FractionalPart[s]; Print[n]], {n, 2, 1378963718}]
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PROGRAM
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(PARI) H(n) = sum(k=1, n, 1/k)+0.; { hr(m)=local(rec); rec=0.5; for(n=2, m, if(frac(H(n))<rec, rec=frac(H(n)); print1(n", "))) }
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CROSSREFS
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Subset of A002387.
Sequence in context: A032110 A054234 A000850 this_sequence A125888 A167418 A055979
Adjacent sequences: A004793 A004794 A004795 this_sequence A004797 A004798 A004799
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KEYWORD
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more,nonn
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
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Edited and extended by Jason Earls (zevi_35711(AT)yahoo.com), Jun 30 2003
Extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 14 2003
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