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A027611 Denominator of n * n-th harmonic number.

%I A027611 #78 Feb 16 2025 08:32:35
%S A027611 1,1,2,3,12,10,20,35,280,252,2520,2310,27720,25740,24024,45045,720720,
%T A027611 680680,4084080,3879876,739024,235144,5173168,14872858,356948592,
%U A027611 343219800,2974571600,2868336900,80313433200,77636318760
%N A027611 Denominator of n * n-th harmonic number.
%C A027611 This is very similar to A128438, which is a different sequence. They differ at n=6 (and nowhere else?). - _N. J. A. Sloane_, Nov 21 2008
%C A027611 Denominator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.
%C A027611 Denominator of Sum_{k=1..n} frac(n/k) where frac(x/y) denotes the fractional part of x/y. - _Benoit Cloitre_, Oct 03 2002
%C A027611 Denominator of Sum_{d=2..n-1, n mod d > 0} n/d. Numerator = A079076. - _Reinhard Zumkeller_, Dec 21 2002
%C A027611 a(n) is odd iff n is a power of 2. - _Benoit Cloitre_, Oct 03 2002
%C A027611 Indices where a(n) differs from A128438 are terms of A074791. - _Gary Detlefs_, Sep 03 2011
%H A027611 Reinhard Zumkeller, <a href="/A027611/b027611.txt">Table of n, a(n) for n = 1..1000</a>
%H A027611 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteSet.html">Complete Set</a>
%F A027611 From _Vladeta Jovovic_, Sep 02 2002: (Start)
%F A027611 a(n) = denominators of coefficients in expansion of -log(1-x)/(1-x)^2.
%F A027611 a(n) = denominators of (n+1)*(harmonic(n+1) - 1).
%F A027611 a(n) = denominators of (n+1)*(Psi(n+2) + Euler-gamma - 1). (End)
%F A027611 a(n) = numerator(h(n)/h(n-1)) - denominator(h(n)/h(n-1)), n > 1, where h(n) is the n-th harmonic number. - _Gary Detlefs_, Sep 03 2011
%F A027611 a(n) = A213999(n, n-2) for n > 1. - _Reinhard Zumkeller_, Jul 03 2012
%F A027611 a(n) = denominators of coefficients of e.g.f. -1 + exp(x)*(1 + Sum_{j >= 0} (-x)^(j+1)/(j * j!)). - _G. C. Greubel_, Aug 24 2022
%p A027611 a := n -> denom(add((n-j)/j, j=1..n));
%p A027611 seq(a(n), n = 1..30); # _Peter Luschny_, May 12 2023
%t A027611 a[n_]:=Denominator[n*HarmonicNumber[n]]; Array[a,100] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2011 *)
%o A027611 (Haskell)
%o A027611 import Data.Ratio ((%), denominator)
%o A027611 a027611 n = denominator $ sum $ map (n %) [1..n]
%o A027611 -- _Reinhard Zumkeller_, Jul 03 2012
%o A027611 (Magma) [Denominator(n*HarmonicNumber(n)): n in [1..40]]; // _Vincenzo Librandi_, Feb 19 2014
%o A027611 (PARI) a(n) = denominator(n*sum(k=1, n, 1/k)); \\ _Michel Marcus_, Feb 15 2015
%o A027611 (Python)
%o A027611 from sympy import harmonic
%o A027611 def A027611(n): return (n*harmonic(n)).q # _Chai Wah Wu_, Sep 26 2021
%o A027611 (SageMath) [denominator(n*harmonic_number(n)) for n in (1..40)] # _G. C. Greubel_, Aug 24 2022
%Y A027611 Harmonic numbers = A001008/A002805.
%Y A027611 Cf. A001705, A006675, A027612, A049820, A024816.
%Y A027611 Cf. A128438, A074791, A079076.
%K A027611 nonn,easy,frac
%O A027611 1,3
%A A027611 Glen Burch (gburch(AT)erols.com)
%E A027611 Entry revised by _N. J. A. Sloane_ following a suggestion of _Eric W. Weisstein_, Jul 02 2004