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

2, 9, 22, 125, 137, 343, 726, 6849, 7129, 81191, 83711, 1118273, 1145993, 1171733, 2391514, 41421503, 42142223, 271211719, 275295799, 55835135, 18858053, 439143531, 1332950097, 33695573875, 34052522467, 309561680403, 312536252003, 9146733078187, 9227046511387

Offset: 2

Author

Gary Detlefs, Aug 04 2011

Keywords

• nonn
• frac

Comments

a(n) mod n^3 = 0 iff n is prime > 3. - Gary Detlefs, Jan 30 2013

Links

• Michael S. Branicky, Table of n, a(n) for n = 2..1001

Crossrefs

Cf. A001008, A002805.

Numerators are in A096617.

Programs

Maple

H:= n-> add(1/k, k=1..n): seq(denom(H(n)/H(n-1)), n=2..25);

Mathematica

h[n_] := Sum[1/i, {i, n}]; Table[Denominator[h[n]/h[n - 1]], {n, 2, 50}] (* T. D. Noe, Aug 04 2011 *)
Denominator[#[[2]]/#[[1]]]&/@Partition[HarmonicNumber[Range[30]],2,1] (* Harvey P. Dale, Jul 05 2015 *)

Python

from fractions import Fraction
def aupton(nn):
  Hnm1, alst = Fraction(1, 1), []
  for n in range(2, nn+1):
    Hn = Hnm1 + Fraction(1, n)
    alst.append((Hn/Hnm1).denominator)
    Hnm1 = Hn
  return alst
print(aupton(30)) # Michael S. Branicky, Feb 09 2021

Formula

a(n) = denominator(H(n)/H(n-1)), where H(n) = Sum_{k=1..n} 1/k.

a(n) = numerator(n*H(n))-denominator(n*H(n)). - Gary Detlefs, Sep 05 2011