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%I #14 May 09 2020 03:36:46
%S 1,1,5,7,47,37,319,533,1879,1627,20417,18107,263111,237371,52279,
%T 95549,1768477,1632341,167324635,155685007,166770367,156188887,
%U 3825136961,3602044091,19081066231,18051406831,57128792093,54260455193
%N 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.
%C 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.
%C 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}.
%C 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).
%C 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.
%F a(n) = abs(numerator(Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)*i/j)).
%e 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
%t Abs[Numerator[Table[Sum[Sum[(-1)^(i+j)*i/j,{i,1,n}],{j,1,n}],{n,1,50}]]]
%o (PARI) a(n) = abs(numerator(sum(j=1, n, sum(i=1, n, (-1)^(i+j)*i/j)))); \\ _Michel Marcus_, May 09 2020
%Y Cf. A058313, A334724 (denominators)
%K frac,nonn
%O 1,3
%A _Alexander Adamchuk_, Jul 12 2006
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