cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A082688 Denominator of Sum_{k=1..n} 1/(n+k).

Original entry on oeis.org

2, 12, 60, 840, 2520, 27720, 360360, 144144, 2450448, 232792560, 232792560, 5354228880, 26771144400, 11473347600, 332727080400, 20629078984800, 20629078984800, 144403552893600, 5342931457063200, 5342931457063200
Offset: 1

Views

Author

Benoit Cloitre, Apr 12 2003

Keywords

Examples

			1/2, 7/12, 37/60, 533/840, 1627/2520, 18107/27720, 237371/360360, ...
		

Crossrefs

Cf. A058312, A082687 (numerators).

Programs

  • Mathematica
    Table[HarmonicNumber[2 n] - HarmonicNumber[n], {n, 20}] // Denominator (* Eric W. Weisstein, Dec 14 2017 *)
  • PARI
    a(n) = denominator(sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

Formula

limit n ->infinity Sum_{k=1..n} 1/(n+k) = log(2).
Denominator of Psi(2*n+1)-Psi(n+1). - Vladeta Jovovic, Aug 24 2003

A250344 Numerator of the harmonic mean of the first n hexagonal numbers.

Original entry on oeis.org

1, 12, 90, 1680, 6300, 83160, 1261260, 576576, 11027016, 1163962800, 1280359080, 32125373280, 174012438600, 80313433200, 2495453103000, 165032631878400, 175347171370800, 1299631976042400, 50757848842100400, 53429314570632000, 2300131992265707600
Offset: 1

Views

Author

Colin Barker, Nov 19 2014

Keywords

Examples

			a(3) = 90 because the first 3 hexagonal numbers are [1,6,15], and 3/(1/1+1/6+1/15) = 90/37.
		

Crossrefs

Cf. A000384 (hexagonal numbers), A082687 (denominators).

Programs

  • PARI
    harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
    s=vector(30); for(n=1, #s, s[n]=numerator(harmonicmean(vector(n, k, k*(2*k-1))))); s

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).

Original entry on oeis.org

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

Views

Author

Alexander Adamchuk, May 15 2007

Keywords

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).

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} ] ].
Previous Showing 11-13 of 13 results.