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 A117731 Numerator of the fraction n*Sum_{k=1..n} 1/(n+k). 9

%I

%S 1,7,37,533,1627,18107,237371,95549,1632341,155685007,156188887,

%T 3602044091,18051406831,54260455193,225175759291,13981692518567,

%U 14000078506967,98115155543129,3634060848592973,3637485804655193

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

%C a(n) almost always equals A082687(n), but differs for n in A125740.

%C p divides a((p-1)/3) for primes p in A002476, that is, primes of form 6*n + 1. - _Alexander Adamchuk_, Jul 16 2006

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.

%F a(n) = numerator(n*Sum_{k=1..n} 1/(n+k)).

%F a(n) = numerator(n*(Psi(2*n+1) - Psi(n+1))).

%F a(n) = numerator(n*Sum_{k=1..2*n} (-1)^(k+1)/k).

%F a(n) = numerator(n*A058313(2*n)/A058312(2*n)).

%F a(n) = numerator(Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1)), which is the numerator of the sum of all matrix elements of n X n Hilbert Matrix M(i,j) = 1/(i+j-1), (i,j = 1..n). The denominator is A117664(n). - _Alexander Adamchuk_, Apr 23 2006

%e The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/A296519.

%e For n=2, the n X n Hilbert matrix is

%e 1 1/2

%e 1/2 1/3

%e Thus, a(2) = numerator(1 + 1/2 + 1/2 + 1/3) = numerator(7/3) = 7.

%e The n X n Hilbert matrix begins as follows:

%e 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...

%e 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...

%e 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...

%e 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...

%e 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...

%e 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

%e ...

%t Numerator[Table[n Sum[1/(n + k), {k, n}], {n, 1, 100}]]

%t Numerator[Table[Sum[Sum[1/(i + j - 1), {i, n}], {j, n}], {n, 30}]] (* _Alexander Adamchuk_, Apr 23 2006 *)

%t Table[n (HarmonicNumber[2 n] - HarmonicNumber[n]), {n, 20}] // Numerator (* _Eric W. Weisstein_, Dec 14 2017 *)

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

%Y Cf. A296519 (denominators).

%Y Cf. A001008, A002476, A005249, A058313, A058312, A082687, A086881, A098118, A117664, A125740.

%K frac,nonn

%O 1,2

%A _Alexander Adamchuk_, Apr 14 2006

%E Various sections edited by _Petros Hadjicostas_ and _Michel Marcus_, May 07 2020

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Last modified April 17 02:26 EDT 2021. Contains 343059 sequences. (Running on oeis4.)