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A120301 Absolute value of numerator of the sum of all matrix elements of n X n matrix M[i,j] = (-1)^(i+j) * i/j, (i,j=1..n). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Up to n=18 a(n) is the same as A058313[n]=numerator of the n-th alternating harmonic number, sum ((-1)^(k+1)/k, k=1..n). 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 = 6k-1, then p divides a(4k-1), otherwise if p = 6k+1, then p divides a(4k). Sum[Sum[(-1)^(i+j)*i/j, {i, 1, n}], {j, 1, n}] = -1/4(2(-1)^n*n+(-1)^n-1) * Sum[(-1)^(k+1)*1/k, {k, 1, n}].

LINKS

Table of n, a(n) for n=1..28.

FORMULA

a(n) = Abs[Numerator[Sum[Sum[(-1)^(i+j)*i/j,{i,1,n}],{j,1,n}]]].

MATHEMATICA

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

CROSSREFS

Cf. A058313.

Sequence in context: A174267 A075830 A058313 * A119787 A025530 A106114

Adjacent sequences:  A120298 A120299 A120300 * A120302 A120303 A120304

KEYWORD

frac,nonn

AUTHOR

Alexander Adamchuk, Jul 12 2006

STATUS

approved

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Last modified February 20 01:26 EST 2018. Contains 299357 sequences. (Running on oeis4.)