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

%I A124838 #24 Feb 16 2025 08:33:03
%S A124838 1,2,6,4,20,10,70,56,504,420,4620,3960,3432,6006,90090,80080,1361360,
%T A124838 408408,369512,67184,470288,1293292,29745716,27457584,228813200,
%U A124838 212469400,5736673800,5354228880,155272637520,291136195350,273491577450
%N A124838 Denominators of third-order harmonic numbers (defined by Conway and Guy, 1996).
%C A124838 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.
%D A124838 J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.
%H A124838 Reinhard Zumkeller, <a href="/A124838/b124838.txt">Table of n, a(n) for n = 1..1000</a>
%H A124838 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 A124838 A124837(n)/A124838(n) = Sum_{i=1..n} A027612(n)/A027611(n+1).
%F A124838 a(n) = denominator(Sum_{m=1..n} Sum_{L=1..m} Sum_{k=1..L} 1/k).
%F A124838 a(n) = denominator(((n+2)!/(2!*n!)) * Sum_{k=3..n+2} 1/k). - _Alexander Adamchuk_, Nov 11 2006
%F A124838 a(n) = A213999(n+2,n-1). - _Reinhard Zumkeller_, Jul 03 2012
%e A124838 a(1) = 1 = denominator of 1/1.
%e A124838 a(2) = 2 = denominator of 1/1 + 5/2 = 7/2.
%e A124838 a(3) = 6 = denominator of 7/2 + 13/3 = 47/6.
%e A124838 a(4) = 4 = denominator of 47/6 + 77/12 = 57/4.
%e A124838 a(5) = 20 = denominator of 57/4 + 87/10 = 549/20.
%e A124838 a(6) = 10 = denominator of 549/20 + 223/20 = 341/10
%e A124838 a(7) = 70 = denominator of 341/10 + 481/35 = 3349/70.
%e A124838 a(8) = 1260 = denominator of 3349/70 + 4609/280 = 88327/1260.
%e A124838 a(9) = 45 = denominator of 88327/1260 + 4861/252 = 3844/45.
%e A124838 a(10) = 504 = denominator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:
%e A124838 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.
%t A124838 Table[Denominator[(n+2)!/2!/n!*Sum[1/k,{k,3,n+2}]],{n,1,40}] (* _Alexander Adamchuk_, Nov 11 2006 *)
%o A124838 (Haskell)
%o A124838 a124838 n = a213999 (n + 2) (n - 1) -- _Reinhard Zumkeller_, Jul 03 2012
%Y A124838 Cf. A027611, A027612, A124837.
%K A124838 easy,frac,nonn
%O A124838 1,2
%A A124838 _Jonathan Vos Post_, Nov 10 2006
%E A124838 Corrected and extended by _Alexander Adamchuk_, Nov 11 2006