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

%I A124837 #28 Feb 16 2025 08:33:03
%S A124837 1,7,47,57,459,341,3349,3601,42131,44441,605453,631193,655217,1355479,
%T A124837 23763863,24444543,476698557,162779395,166474515,34000335,265842403,
%U A124837 812400067,20666950267,21010170067,192066102203,194934439103
%N A124837 Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).
%C A124837 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 A124837 From _Alexander Adamchuk_, Nov 11 2006: (Start)
%C A124837 a(n) is the numerator of H(n, (3)) = Sum_{m=1..n} Sum_{k=1..m} HarmonicNumber(k).
%C A124837 Denominators are listed in A124838.
%C A124837 p divides a(p-5) for prime p > 5.
%C A124837 Primes are listed in A129880.
%C A124837 Numbers k such that a(k) is prime are listed in A129881. (End)
%D A124837 J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.
%H A124837 Reinhard Zumkeller, <a href="/A124837/b124837.txt">Table of n, a(n) for n = 1..1000</a>
%H A124837 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>. See equation for third order harmonic numbers.
%F A124837 A124837(n)/A124838(n) = Sum{i=1..n} A027612(n)/A027611(n+1).
%F A124837 From _Alexander Adamchuk_, Nov 11 2006: (Start)
%F A124837 a(n) = numerator(Sum_{m=1..n} Sum_{l=1..m} Sum_{k=1..l} 1/k).
%F A124837 a(n) = numerator(((n+2)!/(2!*n!)) * Sum_{k=3..n+2} 1/k).
%F A124837 a(n) = numerator(((n+2)*(n+1)/2) * Sum_{k=3..n+2} 1/k). (End)
%F A124837 a(n) = numerator(Sum_{k=0..n-1} (-1)^k*binomial(-3,k)/(n-k)). - _Gary Detlefs_, Jul 18 2011
%F A124837 a(n) = A213998(n+2,n-1). - _Reinhard Zumkeller_, Jul 03 2012
%e A124837 a(1) = 1 = numerator of 1/1.
%e A124837 a(2) = 7 = numerator of 1/1 + 5/2 = 7/2.
%e A124837 a(3) = 47 = numerator of 7/2 + 13/3 = 47/6.
%e A124837 a(4) = 57 = numerator of 47/6 + 77/12 = 57/4.
%e A124837 a(5) = 549 = numerator of 57/4 + 87/10 = 549/20.
%e A124837 a(6) = 341 = numerator of 549/20 + 223/20 = 341/10
%e A124837 a(7) = 3349 = numerator of 341/10 + 481/35 = 3349/70.
%e A124837 a(8) = 88327 = numerator of 3349/70 + 4609/280 = 88327/1260.
%e A124837 a(9) = 3844 = numerator of 88327/1260 + 4861/252 = 3844/45.
%e A124837 a(10) = 54251 = numerator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:
%e A124837 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 A124837 Table[Numerator[(n+2)!/2!/n!*Sum[1/k,{k,3,n+2}]],{n,1,30}] (* _Alexander Adamchuk_, Nov 11 2006 *)
%o A124837 (Haskell)
%o A124837 a124837 n = a213998 (n + 2) (n - 1) -- _Reinhard Zumkeller_, Jul 03 2012
%Y A124837 Cf. A027611, A027612, A124838.
%Y A124837 Cf. A001008, A002805, A067657, A056903, A124878, A124879, A124837, A129880, A129881.
%K A124837 easy,frac,nonn
%O A124837 1,2
%A A124837 _Jonathan Vos Post_, Nov 10 2006
%E A124837 Corrected and extended by _Alexander Adamchuk_, Nov 11 2006