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A296519
Denominator of n*Sum_{k=1..n} 1/(n+k).
2
2, 6, 20, 210, 504, 4620, 51480, 18018, 272272, 23279256, 21162960, 446185740, 2059318800, 5736673800, 22181805360, 1289317436550, 1213475234400, 8022419605200, 281206918792800, 267146572853160, 10431437606647200, 428163098127382800, 409547311252279200
OFFSET
1,1
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
LINKS
FORMULA
From G. C. Greubel, Jul 24 2023: (Start)
a(n) = 2*A117664(n).
a(n) = 2*A111876(n-1)/n. (End)
EXAMPLE
The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/a(n).
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
PROG
(PARI) a(n) = denominator(n*sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017
(Magma) [Denominator(n*(HarmonicNumber(2*n) -HarmonicNumber(n))): n in [1..40]]; // G. C. Greubel, Jul 24 2023
(SageMath) [denominator(n*(harmonic_number(2*n, 1) - harmonic_number(n, 1))) for n in range(1, 41)] # G. C. Greubel, Jul 24 2023
CROSSREFS
Cf. A111876, A117731 (numerators), A117664.
Sequence in context: A274714 A074008 A156334 * A082690 A104861 A074859
KEYWORD
nonn,frac
AUTHOR
Eric W. Weisstein, Dec 14 2017
STATUS
approved