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A124838
Denominators of third-order harmonic numbers (defined by Conway and Guy, 1996).
5
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
OFFSET
1,2
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.
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.
LINKS
Eric Weisstein's World of Mathematics, Harmonic Number. See equation for third-order harmonic numbers.
FORMULA
A124837(n)/A124838(n) = Sum_{i=1..n} A027612(n)/A027611(n+1).
a(n) = denominator(Sum_{m=1..n} Sum_{L=1..m} Sum_{k=1..L} 1/k).
a(n) = denominator(((n+2)!/(2!*n!)) * Sum_{k=3..n+2} 1/k). - Alexander Adamchuk, Nov 11 2006
a(n) = A213999(n+2,n-1). - Reinhard Zumkeller, Jul 03 2012
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.
MATHEMATICA
Table[Denominator[(n+2)!/2!/n!*Sum[1/k, {k, 3, n+2}]], {n, 1, 40}] (* Alexander Adamchuk, Nov 11 2006 *)
PROG
(Haskell)
a124838 n = a213999 (n + 2) (n - 1) -- Reinhard Zumkeller, Jul 03 2012
CROSSREFS
KEYWORD
easy,frac,nonn
AUTHOR
Jonathan Vos Post, Nov 10 2006
EXTENSIONS
Corrected and extended by Alexander Adamchuk, Nov 11 2006
STATUS
approved