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A193758 Denominator of H(n)/H(n-1), where H(n) is the n-th harmonic number = Sum_{k=1..n} 1/k.

%I A193758 #29 Feb 09 2021 19:38:48
%S A193758 2,9,22,125,137,343,726,6849,7129,81191,83711,1118273,1145993,1171733,
%T A193758 2391514,41421503,42142223,271211719,275295799,55835135,18858053,
%U A193758 439143531,1332950097,33695573875,34052522467,309561680403,312536252003,9146733078187,9227046511387
%N A193758 Denominator of H(n)/H(n-1), where H(n) is the n-th harmonic number = Sum_{k=1..n} 1/k.
%C A193758 a(n) mod n^3 = 0 iff n is prime > 3. - _Gary Detlefs_, Jan 30 2013
%H A193758 Michael S. Branicky, <a href="/A193758/b193758.txt">Table of n, a(n) for n = 2..1001</a>
%F A193758 a(n) = denominator(H(n)/H(n-1)), where H(n) = Sum_{k=1..n} 1/k.
%F A193758 a(n) = numerator(n*H(n))-denominator(n*H(n)). - _Gary Detlefs_, Sep 05 2011
%p A193758 H:= n-> add(1/k, k=1..n): seq(denom(H(n)/H(n-1)), n=2..25);
%t A193758 h[n_] := Sum[1/i, {i, n}]; Table[Denominator[h[n]/h[n - 1]], {n, 2, 50}] (* _T. D. Noe_, Aug 04 2011 *)
%t A193758 Denominator[#[[2]]/#[[1]]]&/@Partition[HarmonicNumber[Range[30]],2,1] (* _Harvey P. Dale_, Jul 05 2015 *)
%o A193758 (Python)
%o A193758 from fractions import Fraction
%o A193758 def aupton(nn):
%o A193758   Hnm1, alst = Fraction(1, 1), []
%o A193758   for n in range(2, nn+1):
%o A193758     Hn = Hnm1 + Fraction(1, n)
%o A193758     alst.append((Hn/Hnm1).denominator)
%o A193758     Hnm1 = Hn
%o A193758   return alst
%o A193758 print(aupton(30)) # _Michael S. Branicky_, Feb 09 2021
%Y A193758 Cf. A001008, A002805.
%Y A193758 Numerators are in A096617.
%K A193758 nonn,frac
%O A193758 2,1
%A A193758 _Gary Detlefs_, Aug 04 2011