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A129764 Numerator of the sum of all elements of n X n X n cubic array M[i,j,k] = 1/(i+j+k-2). 0

%I #3 Mar 31 2012 13:20:36

%S 1,15,1133,1177,129149,349673,57087959,345322023,14272692271,

%T 40165727117,217549734472087,14553241481573,18901300532988407,

%U 40603763694792631,9565202506169243753,63888449105310899

%N Numerator of the sum of all elements of n X n X n cubic array M[i,j,k] = 1/(i+j+k-2).

%C a(n) is a 3-d analog of Wolstenholme Numbers (A001008) that are the numerators of Harmonic Numbers H(n) = Sum[ 1/i, {i,1,n} ]. n X n X n cubic array M[i,j,k] = 1/(i+j+k-2) is a 3-d analog of n X n Hilbert Matrix with elements M[i,j] = 1/(i+j-1). p divides a((p+1)/3) for prime p = {5,11,17,23,29,41,47,53,59,71,83,89,...} = A007528 Primes of form 6n-1. Sum[ Sum[ Sum[ (i+j+k-2), {i,1,n} ], {j,1,n} ], {k,1,n} ] = 1/2*n^3*(3n-1).

%H Eric Weisstein, The World of Mathematics: <a href="http://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.

%H Eric Weisstein, The World of Mathematics: <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.

%F a(n) = Numerator[ Sum[ Sum[ Sum[ 1/(i+j+k-2), {i,1,n} ], {j,1,n} ], {k,1,n} ] ].

%t Table[ Numerator[ Sum[ Sum[ Sum[ 1/(i+j+k-2), {i,1,n} ], {j,1,n} ], {k,1,n} ] ], {n,1,30} ]

%Y Cf. A001008 = Wolstenholme numbers: numerator of harmonic number H(n)=Sum_{i=1..n} 1/i. Cf. A082687, A117731, A007528.

%K nonn

%O 1,2

%A _Alexander Adamchuk_, May 15 2007

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