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A124838
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Denominators of third order harmonic numbers (defined by Conway and Guy, 1996).
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2
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1, 2, 6, 4, 20, 10, 70, 56, 504, 420, 4620, 3960, 3432, 6006, 90090, 80080, 1361360, 408408, 369512, 67184, 470288, 1293292, 29745716, 27457584, 228813200, 212469400, 5736673800, 5354228880, 155272637520, 291136195350, 273491577450
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Numerators are A124837. All fractions reduced. Thanks to Jonathan Sondow for verifying these calculations. He suggests that the equivalent definition in terms of first order harmonic numbers may be computationally simpler. We are happy with the description of A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n, but baffled by the description of A027611.
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REFERENCES
| J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.
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LINKS
| Jonathan Sondow and Eric Weisstein's World of Mathematics, Harmonic Number. See equation for third order harmonic numbers.
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FORMULA
| A124837(n)/A124838(n) = SUM[i=1..n] A027612(n)/A027611(n+1).
a(n) = Denominator[ Sum[ Sum[ Sum[ 1/k, {k,1,l} ], {l,1,m} ], {m,1,n} ] ]. a(n) = Denominator[ (n+2)!/2!/n! * Sum[ 1/k, {k,3,n+2} ] ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 11 2006
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EXAMPLE
| a(1) = 1 = denominator of 1/1.
a(2) = 2 = denominator of 1/1 + 5/2 = 7/2.
a(3) = 6 = denominator of 7/2 + 13/3 = 47/6.
a(4) = 4 = denominator of 47/6 + 77/12 = 57/4.
a(5) = 20 = denominator of 57/4 + 87/10 = 549/20.
a(6) = 10 = denominator of 549/20 + 223/20 = 341/10
a(7) = 70 = denominator of 341/10 + 481/35 = 3349/70.
a(8) = 1260 = denominator of 3349/70 + 4609/280 = 88327/1260.
a(9) = 45 = denominator of 88327/1260 + 4861/252 = 3844/45.
a(10) = 504 = denominator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:
a(10) = 504 = denominator of 1/1 + 5/2 + 13/3 + 77/12 + 87/10 + 223/20 + 481/35 + 4609/252 + 4861/252 + 55991/2520 = 54251/504.
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MATHEMATICA
| Table[Denominator[(n+2)!/2!/n!*Sum[1/k, {k, 3, n+2}]], {n, 1, 40}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 11 2006
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CROSSREFS
| Cf. A027611, A027612, A124837.
Sequence in context: A174824 A009262 A127699 * A088659 A052100 A079579
Adjacent sequences: A124835 A124836 A124837 * A124839 A124840 A124841
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KEYWORD
| easy,frac,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 10 2006
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EXTENSIONS
| Corrected and extended by Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 11 2006
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