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.

Showing 1-3 of 3 results.

A335130 a(n) = 1 + Sum_{k=1..n} sign((sign(n+Sum_{j=2..k}-|A334312(n,j)|)+1)).

Original entry on oeis.org

2, 3, 4, 5, 6, 5, 7, 7, 10, 7, 11, 10, 11, 10, 11, 11, 14, 13, 14, 13, 13, 11, 15, 12, 17, 13, 17, 13, 19, 13, 14, 15, 15, 15, 17, 17, 17, 19, 19, 17, 19, 17, 18, 17, 17, 18, 21, 19, 22, 21, 21, 19, 21, 21, 23, 23, 19, 19, 26, 19, 19, 23, 23, 23, 29, 21, 22, 21, 23, 21, 23, 22, 23, 23, 28, 23, 31, 23, 26, 25
Offset: 1

Views

Author

Mats Granvik, May 24 2020

Keywords

Comments

a(n) appears to be asymptotic to sqrt(8*n).

Crossrefs

Cf. A334312.

Programs

  • Mathematica
    nn = 80; varphi[n_] := DivisorSum[n, MoebiusMu[#] # &]; A = Table[Table[Sum[If[n >= k, varphi[GCD[i, k]], 0], {i, k, n}], {k, 1, nn}], {n, 1, nn}]; vv = Table[1 + Sum[Sign[(1 + Sign[Sum[If[j == 1, A[[n, j]], -Abs[A[[n, j]]]], {j, 1, k}]])], {k, 1, n}], {n, 1, nn}]

Formula

a(n) = 1 + Sum_{k=1..n} sign((sign(n+Sum_{j=2..k}-|A334312(n,j)|)+1)).

A334313 Denominators of the partial sums of the Möbius transform of the harmonic numbers.

Original entry on oeis.org

1, 2, 3, 12, 5, 60, 210, 840, 63, 2520, 770, 5544, 25740, 40040, 90090, 720720, 510510, 12252240, 58198140, 232792560, 5290740, 3695120, 148728580, 5354228880, 147094200, 26771144400, 5019589575, 80313433200, 2587877292, 2329089562800, 36100888223400, 3702655202400
Offset: 1

Views

Author

Mats Granvik, Apr 23 2020

Keywords

Crossrefs

Numerators are in A334314.
Cf. A334312.

Programs

  • Mathematica
    nn = 32; Denominator[Table[Sum[Sum[If[Mod[n, k] == 0, MoebiusMu[n/k]*HarmonicNumber[k], 0], {k, 1, n}], {n, 1, m}], {m, 1, nn}]]
  • PARI
    a(n) = denominator(sum(m=1, n, sumdiv(m, d, moebius(m/d)*sum(i=1, d, 1/i)))); \\ Michel Marcus, Apr 23 2020

Formula

a(n) = denominator of Sum_{m=1..n} Sum_{d|m} H(d)*mu(m/d).
A334314(n)/a(n) = Sum_{k=1..n} A334312(n,k)/k.

A334314 Numerators of the partial sums of the Möbius transform of the harmonic numbers.

Original entry on oeis.org

1, 3, 7, 35, 21, 259, 1241, 5497, 475, 19367, 7473, 54193, 307727, 485041, 1109501, 9353753, 7870991, 189509941, 1048445929, 4213673063, 96924859, 68325181, 3156755829, 113847868229, 3353128913, 614935296797, 120633624344, 1937548941997, 70096529185, 62887637910847
Offset: 1

Views

Author

Mats Granvik, Apr 22 2020

Keywords

Crossrefs

Denominators are in A334313.
Cf. A001008 (numerators of harmonic numbers), A334312.

Programs

  • Mathematica
    nn = 30; Numerator[Table[Sum[Sum[If[Mod[n, k] == 0, MoebiusMu[n/k]*HarmonicNumber[k], 0], {k, 1, n}], {n, 1, m}], {m, 1, nn}]]
  • PARI
    a(n) = numerator(sum(m=1, n, sumdiv(m, d, moebius(m/d)*sum(i=1, d, 1/i)))); \\ Michel Marcus, Apr 23 2020

Formula

a(n) = numerator of Sum_{m=1..n} Sum_{d|m} H(d)*mu(m/d).
a(n)/A334313(n) = Sum_{k=1..n} A334312(n,k)/k.
Showing 1-3 of 3 results.