cp's OEIS Frontend

A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.

%I A027612 #50 Feb 16 2025 08:32:35
%S A027612 1,5,13,77,87,223,481,4609,4861,55991,58301,785633,811373,835397,
%T A027612 1715839,29889983,30570663,197698279,201578155,41054655,13920029,
%U A027612 325333835,990874363,25128807667,25472027467,232222818803,235091155703,6897956948587,6975593267347
%N A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n.
%C A027612 Numerator of a second-order harmonic number H(n, (2)) = Sum_{k=1..n} HarmonicNumber(k). - _Alexander Adamchuk_, Apr 12 2006
%C A027612 p divides a(p-3) for prime p > 3. - _Alexander Adamchuk_, Jul 06 2006
%C A027612 Denominator is A027611(n+1). p divides a(p-3) for prime p > 3. - _Alexander Adamchuk_, Jul 26 2006
%C A027612 a(n) = A213998(n,n-2) for n > 1. - _Reinhard Zumkeller_, Jul 03 2012
%H A027612 Reinhard Zumkeller, <a href="/A027612/b027612.txt">Table of n, a(n) for n = 1..1000</a>
%H A027612 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%F A027612 From _Vladeta Jovovic_, Sep 02 2002: (Start)
%F A027612 a(n) = numerators of coefficients in expansion of -log(1-x)/(1-x)^2.
%F A027612 a(n) = numerators of (n+1)*(harmonic(n+1) - 1).
%F A027612 a(n) = numerators of (n+1)*(Psi(n+2) + Euler-gamma - 1). (End)
%F A027612 a(n) = numerator( Sum_{k=1..n} Sum_{i=1..k} 1/i ). - _Alexander Adamchuk_, Apr 12 2006
%F A027612 a(n) = numerator( Sum_{k=1..n} k/(n-k+1) ). - _Alexander Adamchuk_, Jul 26 2006
%F A027612 a(n) = numerator of integral_{x=1..n+1} floor((n+1)/x). - _Jean-François Alcover_, Jun 18 2013
%p A027612 a := n -> numer(add((n+1-j)/j, j=1..n));
%p A027612 seq(a(n), n = 1..29); # _Peter Luschny_, May 12 2023
%t A027612 Numerator[Table[Sum[Sum[1/i,{i,1,k}],{k,1,n}],{n,1,30}]] (* _Alexander Adamchuk_, Apr 12 2006 *)
%t A027612 Numerator[Table[Sum[k/(n-k+1),{k,1,n}],{n,1,50}]] (* _Alexander Adamchuk_, Jul 26 2006 *)
%o A027612 (Haskell)
%o A027612 import Data.Ratio ((%), numerator)
%o A027612 a027612 n = numerator $ sum $ zipWith (%) [1 .. n] [n, n-1 .. 1]
%o A027612 -- _Reinhard Zumkeller_, Jul 03 2012
%o A027612 (PARI) a(n) = numerator(sum(k=1, n, k/(n-k+1))); \\ _Michel Marcus_, Jul 14 2018
%o A027612 (Magma) [Numerator((&+[j/(n-j+1): j in [1..n]])): n in [1..30]]; // _G. C. Greubel_, Aug 23 2022
%o A027612 (SageMath) [numerator(n*(harmonic_number(n+1) - 1)) for n in (1..30)] # _G. C. Greubel_, Aug 23 2022
%Y A027612 Cf. A001008, A001705, A002805, A006675, A022819, A027611, A093418.
%K A027612 nonn,easy,frac
%O A027612 1,2
%A A027612 Glen Burch (gburch(AT)erols.com)