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Hint to puzzle 34: Harmonic sum 2

Consider the table below, which shows Hn, for n = 2 to 10.  The denominator is equal to the least common multiple of 2, 3, ..., n; and is therefore always even.  Notice that, in each case, the numerator is odd.  Is this true for all n > 1?

Harmonic sum
nHn
23/2
311/6
425/12
5137/60
6147/60
71089/420
82283/840
97129/2520
10 7381/2520