A124838 Denominators of third-order harmonic numbers (defined by Conway and Guy, 1996).

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

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

Cf. A027611, A027612, A124837.

Sequence in context: A127699 A100140 A220769 * A247578 A088659 A299822

Adjacent sequences: A124835 A124836 A124837 * A124839 A124840 A124841

Keyword

easy,frac,nonn

Author

Jonathan Vos Post, Nov 10 2006

Extensions

Corrected and extended by Alexander Adamchuk, Nov 11 2006

Status

approved