A296519 - OEIS

%I #21 Jul 24 2023 02:38:02

%S 2,6,20,210,504,4620,51480,18018,272272,23279256,21162960,446185740,

%T 2059318800,5736673800,22181805360,1289317436550,1213475234400,

%U 8022419605200,281206918792800,267146572853160,10431437606647200,428163098127382800,409547311252279200

%N Denominator of n*Sum_{k=1..n} 1/(n+k).

%C a(n) is divisible by all primes p such that the numerator of Sum_{n < k*p <= n} 1/k is not divisible by p, in particular by all primes from n+1 to 2*n-1. - _Robert Israel_, May 21 2020

%H Robert Israel, <a href="/A296519/b296519.txt">Table of n, a(n) for n = 1..1155</a>

%F From _G. C. Greubel_, Jul 24 2023: (Start)

%F a(n) = 2*A117664(n).

%F a(n) = 2*A111876(n-1)/n. (End)

%e The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/a(n).

%p N:= 30: # for a(1)..a(N)

%p H:= ListTools:-PartialSums([seq(1/i,i=1..2*N)]):

%p map(n -> denom(n*(H[2*n]-H[n])), [$1..N]); # _Robert Israel_, May 21 2020

%t Table[n (HarmonicNumber[2 n] - HarmonicNumber[n]), {n, 30}] // Denominator

%o (PARI) a(n) = denominator(n*sum(k=1, n, 1/(n+k))); \\ _Michel Marcus_, Dec 14 2017

%o (Magma) [Denominator(n*(HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // _G. C. Greubel_, Jul 24 2023

%o (SageMath) [denominator(n*(harmonic_number(2*n,1) - harmonic_number(n,1))) for n in range(1,41)] # _G. C. Greubel_, Jul 24 2023

%Y Cf. A111876, A117731 (numerators), A117664.

%K nonn,frac

%O 1,1

%A _Eric W. Weisstein_, Dec 14 2017