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A120301
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Absolute value of numerator of the sum of all elements of the n X n matrix M with M[i,j] = (-1)^(i+j)*i/j for i,j = 1..n.
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4
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1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 52279, 95549, 1768477, 1632341, 167324635, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 54260455193
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OFFSET
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1,3
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COMMENTS
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Up to n = 18, a(n) is the same as A058313(n) = numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k. a(n) differs from A058313(n) only for n = 18, 28, 87, 99.
Up to n = 100 the ratio a(n)/A058313(n) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1}.
A Wolstenholme-like theorem: for prime p > 3, if p = 6*k - 1, then p divides a(4*k-1), otherwise if p = 6*k + 1, then p divides a(4*k).
Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)*i/j = -1/4 * (2*(-1)^n*n + (-1)^n - 1) * Sum_{k=1..n} (-1)^(k+1)/k.
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LINKS
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FORMULA
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a(n) = abs(numerator(Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)*i/j)).
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EXAMPLE
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The absolute values of the first few fractions are 1, 1/2, 5/3, 7/6, 47/20, 37/20, 319/105, 533/210, 1879/504, ... = A120301/A334724. - Petros Hadjicostas, May 09 2020
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MATHEMATICA
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Abs[Numerator[Table[Sum[Sum[(-1)^(i+j)*i/j, {i, 1, n}], {j, 1, n}], {n, 1, 50}]]]
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PROG
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(PARI) a(n) = abs(numerator(sum(j=1, n, sum(i=1, n, (-1)^(i+j)*i/j)))); \\ Michel Marcus, May 09 2020
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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