A124837 Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).

1, 7, 47, 57, 459, 341, 3349, 3601, 42131, 44441, 605453, 631193, 655217, 1355479, 23763863, 24444543, 476698557, 162779395, 166474515, 34000335, 265842403, 812400067, 20666950267, 21010170067, 192066102203, 194934439103

Offset: 1

Author

Jonathan Vos Post, Nov 10 2006

Keywords

• easy
• frac
• nonn

Comments

Denominators are A124838. 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.

From Alexander Adamchuk, Nov 11 2006: (Start)

a(n) is the numerator of H(n, (3)) = Sum_{m=1..n} Sum_{k=1..m} HarmonicNumber(k).

Denominators are listed in A124838.

p divides a(p-5) for prime p > 5.

Primes are listed in A129880.

Numbers k such that a(k) is prime are listed in A129881. (End)

Examples

			a(1) = 1 = numerator of 1/1.
a(2) = 7 = numerator of 1/1 + 5/2 = 7/2.
a(3) = 47 = numerator of 7/2 + 13/3 = 47/6.
a(4) = 57 = numerator of 47/6 + 77/12 = 57/4.
a(5) = 549 = numerator of 57/4 + 87/10 = 549/20.
a(6) = 341 = numerator of 549/20 + 223/20 = 341/10
a(7) = 3349 = numerator of 341/10 + 481/35 = 3349/70.
a(8) = 88327 = numerator of 3349/70 + 4609/280 = 88327/1260.
a(9) = 3844 = numerator of 88327/1260 + 4861/252 = 3844/45.
a(10) = 54251 = numerator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:
a(10) = 54251 = numerator of 1/1 + 5/2 + 13/3 + 77/12 + 87/10 + 223/20 + 481/35 + 4609/252 + 4861/252 + 55991/2520 = 54251/504.
		

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.

Crossrefs

Cf. A027611, A027612, A124838.

Cf. A001008, A002805, A067657, A056903, A124878, A124879, A124837, A129880, A129881.

Programs

Haskell

a124837 n = a213998 (n + 2) (n - 1) -- Reinhard Zumkeller, Jul 03 2012

Mathematica

Table[Numerator[(n+2)!/2!/n!*Sum[1/k,{k,3,n+2}]],{n,1,30}] (* Alexander Adamchuk, Nov 11 2006 *)

Formula

A124837(n)/A124838(n) = Sum{i=1..n} A027612(n)/A027611(n+1).

From Alexander Adamchuk, Nov 11 2006: (Start)

a(n) = numerator(Sum_{m=1..n} Sum_{l=1..m} Sum_{k=1..l} 1/k).

a(n) = numerator(((n+2)!/(2!*n!)) * Sum_{k=3..n+2} 1/k).

a(n) = numerator(((n+2)*(n+1)/2) * Sum_{k=3..n+2} 1/k). (End)

a(n) = numerator(Sum_{k=0..n-1} (-1)^k*binomial(-3,k)/(n-k)). - Gary Detlefs, Jul 18 2011

a(n) = A213998(n+2,n-1). - Reinhard Zumkeller, Jul 03 2012

Extensions

Corrected and extended by Alexander Adamchuk, Nov 11 2006