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A117731
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Numerator of the fraction n*Sum_{k=1..n} 1/(n+k).
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9
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1, 7, 37, 533, 1627, 18107, 237371, 95549, 1632341, 155685007, 156188887, 3602044091, 18051406831, 54260455193, 225175759291, 13981692518567, 14000078506967, 98115155543129, 3634060848592973, 3637485804655193
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OFFSET
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1,2
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COMMENTS
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a(n) almost always equals A082687(n), but differs for n in A125740.
p divides a((p-1)/3) for primes p in A002476, that is, primes of form 6*n + 1. - Alexander Adamchuk, Jul 16 2006
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LINKS
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Table of n, a(n) for n=1..20.
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Hilbert Matrix.
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FORMULA
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a(n) = numerator(n*Sum_{k=1..n} 1/(n+k)).
a(n) = numerator(n*(Psi(2*n+1) - Psi(n+1))).
a(n) = numerator(n*Sum_{k=1..2*n} (-1)^(k+1)/k).
a(n) = numerator(n*A058313(2*n)/A058312(2*n)).
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
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EXAMPLE
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The first few fractions are 1/2, 7/6, 37/20, 533/210, 1627/504, 18107/4620, 237371/51480, ... = A117731/A296519.
For n=2, the n X n Hilbert matrix is
1 1/2
1/2 1/3
Thus, a(2) = numerator(1 + 1/2 + 1/2 + 1/3) = numerator(7/3) = 7.
The n X n Hilbert matrix begins as follows:
1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
...
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MATHEMATICA
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Numerator[Table[n Sum[1/(n + k), {k, n}], {n, 1, 100}]]
Numerator[Table[Sum[Sum[1/(i + j - 1), {i, n}], {j, n}], {n, 30}]] (* Alexander Adamchuk, Apr 23 2006 *)
Table[n (HarmonicNumber[2 n] - HarmonicNumber[n]), {n, 20}] // Numerator (* Eric W. Weisstein, Dec 14 2017 *)
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PROG
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(PARI) a(n) = numerator(n*sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017
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CROSSREFS
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Cf. A296519 (denominators).
Cf. A001008, A002476, A005249, A058313, A058312, A082687, A086881, A098118, A117664, A125740.
Sequence in context: A097493 A341377 A082687 * A155010 A292807 A210620
Adjacent sequences: A117728 A117729 A117730 * A117732 A117733 A117734
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KEYWORD
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frac,nonn
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AUTHOR
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Alexander Adamchuk, Apr 14 2006
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EXTENSIONS
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Various sections edited by Petros Hadjicostas and Michel Marcus, May 07 2020
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STATUS
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approved
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