A124837 - OEIS

%I #27 Aug 03 2023 02:07:10

%S 1,7,47,57,459,341,3349,3601,42131,44441,605453,631193,655217,1355479,

%T 23763863,24444543,476698557,162779395,166474515,34000335,265842403,

%U 812400067,20666950267,21010170067,192066102203,194934439103

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

%C 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.

%C From _Alexander Adamchuk_, Nov 11 2006: (Start)

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

%C Denominators are listed in A124838.

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

%C Primes are listed in A129880.

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

%D J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.

%H Reinhard Zumkeller, <a href="/A124837/b124837.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>. See equation for third order harmonic numbers.

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

%F From _Alexander Adamchuk_, Nov 11 2006: (Start)

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

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

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

%F a(n) = numerator(Sum_{k=0..n-1} (-1)^k*binomial(-3,k)/(n-k)). - _Gary Detlefs_, Jul 18 2011

%F a(n) = A213998(n+2,n-1). - _Reinhard Zumkeller_, Jul 03 2012

%e a(1) = 1 = numerator of 1/1.

%e a(2) = 7 = numerator of 1/1 + 5/2 = 7/2.

%e a(3) = 47 = numerator of 7/2 + 13/3 = 47/6.

%e a(4) = 57 = numerator of 47/6 + 77/12 = 57/4.

%e a(5) = 549 = numerator of 57/4 + 87/10 = 549/20.

%e a(6) = 341 = numerator of 549/20 + 223/20 = 341/10

%e a(7) = 3349 = numerator of 341/10 + 481/35 = 3349/70.

%e a(8) = 88327 = numerator of 3349/70 + 4609/280 = 88327/1260.

%e a(9) = 3844 = numerator of 88327/1260 + 4861/252 = 3844/45.

%e a(10) = 54251 = numerator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:

%e 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.

%t Table[Numerator[(n+2)!/2!/n!*Sum[1/k,{k,3,n+2}]],{n,1,30}] (* _Alexander Adamchuk_, Nov 11 2006 *)

%o (Haskell)

%o a124837 n = a213998 (n + 2) (n - 1) -- _Reinhard Zumkeller_, Jul 03 2012

%Y Cf. A027611, A027612, A124838.

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

%K easy,frac,nonn

%O 1,2

%A _Jonathan Vos Post_, Nov 10 2006

%E Corrected and extended by _Alexander Adamchuk_, Nov 11 2006