A120301 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.

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

Offset: 1

Author

Alexander Adamchuk, Jul 12 2006

Keywords

• frac
• nonn

Comments

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.

Examples

			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
		

Crossrefs

Cf. A058313, A334724 (denominators)

Programs

Mathematica

Abs[Numerator[Table[Sum[Sum[(-1)^(i+j)*i/j,{i,1,n}],{j,1,n}],{n,1,50}]]]

PARI

a(n) = abs(numerator(sum(j=1, n, sum(i=1, n, (-1)^(i+j)*i/j)))); \\ Michel Marcus, May 09 2020

Formula

a(n) = abs(numerator(Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)*i/j)).