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 1-4 of 4 results.

A335316 Harmonic numbers (A001599) with a record harmonic mean of divisors.

Original entry on oeis.org

1, 6, 28, 140, 270, 672, 1638, 2970, 8190, 27846, 30240, 167400, 237510, 332640, 695520, 1421280, 2178540, 2457000, 11981970, 14303520, 17428320, 23963940, 27027000, 46683000, 56511000, 71253000, 142990848, 163390500, 164989440, 191711520, 400851360, 407386980
Offset: 1

Views

Author

Amiram Eldar, May 31 2020

Keywords

Comments

The corresponding record values are 1, 2, 3, 5, 6, 8, 9, 11, 15, ... (see the link for more values).
The terms 1, 6, 30240 and 332640 are also terms of A179971.

Examples

			The first 7 harmonic numbers are 1, 6, 28, 140, 270, 496 and 672. Their harmonic means of divisors (A001600) are 1, 2, 3, 5, 6, 5 and 8. The record values, 1, 2, 3, 5, 6 and 8 occur at 1, 6, 28, 140, 270 and 672, the first 6 terms of this sequence.

Links

Crossrefs

Cf. A001599, A001600, A099377, A099378, A179971, A335317, A335318.

Programs

  • Mathematica
    h[n_] := n * DivisorSigma[0, n] / DivisorSigma[1, n]; hm = 0; s = {}; Do[h1 = h[n];  If[IntegerQ[h1] && h1 > hm, hm = h1; AppendTo[s, n]], {n, 1, 10^6}]; s

A348654 Indices of records in the sequence of unitary harmonic means A103339(k)/A103340(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 28, 30, 42, 60, 84, 105, 120, 140, 180, 210, 330, 390, 420, 660, 780, 840, 1092, 1155, 1260, 1540, 1820, 1980, 2310, 2730, 3570, 3990, 4290, 4620, 5460, 7140, 7980, 8580, 9240, 10920, 13860, 16380, 20020, 23940, 25740, 27720, 30030
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2021

Keywords

Examples

			The unitary harmonic means of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. The next record, A103339(10)/A103340(10) = 20/9, occurs at 10. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 10.

Links

Crossrefs

The unitary version of A179971.
Cf. A103339, A103340.

Programs

  • Mathematica
    f[p_, e_] := 2/(1 + p^(-e)); uhmeam[n_] := Times @@ f @@@ FactorInteger[n]; s = {}; max = 0; Do[u1 = uhmeam[n]; If[u1 > max, max = u1; AppendTo[s, n]], {n, 1, 10^5}]; s

A361319 Indices of records in the sequence of infinitary harmonic means A361316(k)/A361317(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 54, 56, 60, 84, 105, 120, 168, 210, 264, 270, 280, 360, 420, 540, 660, 756, 840, 1080, 1320, 1512, 1848, 1890, 2310, 2520, 3080, 3640, 3780, 4620, 5460, 5940, 7020, 7560, 9240, 10920, 11880, 14040, 16632, 19656
Offset: 1

Views

Author

Amiram Eldar, Mar 09 2023

Keywords

Examples

			The infinitary harmonic means of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. The next record, A361316(8)/A361317(8) = 32/15, occurs at 8. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 8.

Links

Crossrefs

Cf. A361316, A361317.
Similar sequences: A179971, A348654.
Other sequences related to records of infinitary divisors: A037992, A327634.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; ihmean[1] = 1; ihmean[n_] := n*Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{ih, ihmax = 0, s = {}}, Do[ih = ihmean[k]; If[ih > ihmax, ihmax = ih; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[20000]
  • PARI
    ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih, ihmax=0); for(k = 1, kmax, ih = ihmean(k); if(ih > ihmax, ihmax = ih; print1(k, ", ")));}

A361785 Indices of records in the sequence of bi-unitary harmonic means A361782(k)/A361783(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 54, 56, 60, 84, 96, 120, 168, 210, 240, 270, 280, 360, 420, 480, 672, 840, 1080, 1320, 1512, 1680, 1890, 2160, 2310, 2520, 3080, 3360, 4320, 5280, 6048, 7392, 7560, 9240, 10920, 11880, 14040, 15120, 18480, 20790
Offset: 1

Views

Author

Amiram Eldar, Mar 24 2023

Keywords

Examples

			The harmonic means of the bi-unitary divisors of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. A361782(7)/A361783(7) = 9/5 < 2, and the next record, A361782(8)/A361783(8) = 32/15, occurs at 8. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 8.

Links

Crossrefs

Cf. A361782, A361783.
Similar sequences: A179971, A348654, A361319.
Other sequences related to records of bi-unitary divisors: A293185, A292983, A292984.

Programs

  • Mathematica
    f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; buhmean[1] = 1; buhmean[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{buh, buhmax = 0, s = {}}, Do[buh = buhmean[k]; If[buh > buhmax, buhmax = buh; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[20000]
  • PARI
    buhmean(n) = {my(f = factor(n), p, e); n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2)))); }
lista(kmax) = {my(buh, buhmax=0); for(k = 1, kmax, buh = buhmean(k); if(buh > buhmax, buhmax = buh; print1(k, ", "))); }
Showing 1-4 of 4 results.

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