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%I #18 Sep 24 2018 16:53:14
%S 1,9,11,125,137,1029,363,6849,7129,81191,83711,1118273,1145993,
%T 1171733,1195757,41421503,42142223,813635157,275295799,279175675,
%U 56574159,439143531,1332950097,33695573875,34052522467,309561680403,312536252003,9146733078187
%N Numerator of the product of the n-th triangular number and the n-th harmonic number.
%C Also numerator of the sum of all matrix elements of n X n matrix M[i,j] = i/j, i,j=1..n.
%C p^3 divides a(p-1) for prime p>3, p^3 divides a(p^2-1) for prime p>3, p^3 divides a(p^3-1) for prime p>3, p^3 divides a(p^4-1) for prime p>3, ...
%H Alois P. Heinz, <a href="/A119786/b119786.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = numerator[Sum[i,{i, 1, n}] * Sum[1/j,{j, 1, n}]] = numerator[n(n+1)/2 * Sum[1/i,{i, 1, n}]] = numerator[A000217(n) * (A001008(n)/A002805(n))]. Also a(n) = numerator[Sum[Sum[i/j,{i, 1, n}],{j, 1, n}]].
%p a:= n-> numer(add(add(i/j, j=1..n), i=1..n)): seq(a(n), n=1..30); # _Zerinvary Lajos_, Jun 14 2007
%p # second Maple program:
%p h:= proc(n) h(n):= 1/n +`if`(n=1, 0, h(n-1)) end:
%p t:= proc(n) t(n):= n +`if`(n=1, 0, t(n-1)) end:
%p a:= n-> numer(h(n)*t(n)):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, May 24 2013
%t Numerator[Table[n(n+1)/2*Sum[1/i,{i, 1, n}],{n,1,50}]]. Numerator[Table[Sum[Sum[i/j,{i, 1, n}],{j, 1, n}],{n,1,50}]].
%t Table[(n(n+1))/2 HarmonicNumber[n],{n,30}]//Numerator (* _Harvey P. Dale_, May 06 2018 *)
%Y Cf. A000217, A001008, A002805.
%K frac,nonn
%O 1,2
%A _Alexander Adamchuk_, Jun 25 2006, Jul 12 2006
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, May 27 2007
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