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).

1, 15, 1133, 1177, 129149, 349673, 57087959, 345322023, 14272692271, 40165727117, 217549734472087, 14553241481573, 18901300532988407, 40603763694792631, 9565202506169243753, 63888449105310899

Offset: 1

Author

Alexander Adamchuk, May 15 2007

Keywords

• nonn

Comments

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).

Links

• Eric Weisstein, The World of Mathematics: Hilbert Matrix.
• Eric Weisstein, The World of Mathematics: Harmonic Number.

Crossrefs

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

Programs

Mathematica

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

Formula

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