A296519 Denominator of n*Sum_{k=1..n} 1/(n+k).

2, 6, 20, 210, 504, 4620, 51480, 18018, 272272, 23279256, 21162960, 446185740, 2059318800, 5736673800, 22181805360, 1289317436550, 1213475234400, 8022419605200, 281206918792800, 267146572853160, 10431437606647200, 428163098127382800, 409547311252279200

Offset: 1

Author

Eric W. Weisstein, Dec 14 2017

Keywords

• nonn
• frac

Comments

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..1155

Crossrefs

Cf. A111876, A117731 (numerators), A117664.

Programs

Magma

[Denominator(n*(HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // G. C. Greubel, Jul 24 2023

Maple

N:= 30: # for a(1)..a(N)
H:= ListTools:-PartialSums([seq(1/i,i=1..2*N)]):
map(n -> denom(n*(H[2*n]-H[n])), [$1..N]); # Robert Israel, May 21 2020

Mathematica

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

PARI

a(n) = denominator(n*sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

SageMath

[denominator(n*(harmonic_number(2*n,1) - harmonic_number(n,1))) for n in range(1,41)] # G. C. Greubel, Jul 24 2023

Formula

From G. C. Greubel, Jul 24 2023: (Start)

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

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