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.

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, ", ")));}

A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 128, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 256, 65, 11, 67, 136
Offset: 1

Views

Author

Amiram Eldar, Mar 09 2023

Keywords

Examples

			Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...

Links

Crossrefs

Cf. A036537, A037445, A049417, A077609, A063947, A138302, A361317 (denominators).
Similar sequences: A099377, A103339.

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}]]; a[1] = 1; a[n_] := Numerator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), b); numerator(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)))); }

Formula

a(n) = numerator(n*A037445(n)/A049417(n)).
a(n)/A361317(n) <= A099377(n)/A099378(n), with equality if and only if n is in A036537.
a(n)/A361317(n) >= A103339(n)/A103340(n), with equality if and only if n is in A138302.

A361783 Denominators of the harmonic means of the bi-unitary divisors of the positive integers.

Original entry on oeis.org

1, 3, 2, 5, 3, 1, 4, 15, 5, 9, 6, 5, 7, 3, 2, 27, 9, 5, 10, 3, 8, 9, 12, 5, 13, 21, 10, 5, 15, 3, 16, 21, 4, 27, 12, 25, 19, 15, 14, 9, 21, 2, 22, 15, 1, 9, 24, 9, 25, 39, 6, 35, 27, 5, 18, 15, 20, 45, 30, 1, 31, 12, 20, 119, 21, 3, 34, 45, 8, 9, 36, 25, 37, 57
Offset: 1

Views

Author

Amiram Eldar, Mar 24 2023

Keywords

Links

Crossrefs

Cf. A188999, A222266, A286324, A286325 (positions of 1's), A361782 (numerators).
Similar sequences: A099378, A103340, A361317.

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))]; a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), p, e); denominator(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))))); }

Formula

a(n) = denominator(n*A286324(n)/A188999(n)).

A361318 Harmonic means of the infinitary divisors of the infinitary harmonic numbers.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 7, 11, 13, 13, 10, 7, 15, 16, 15, 9, 20, 18, 14, 25, 24, 19, 25, 15, 27, 28, 30, 18, 36, 13, 21, 17, 29, 40, 33, 24, 28, 38, 31, 29, 45, 34, 27, 28, 44, 27, 60, 36, 52, 46, 26, 51, 42, 55, 33, 66, 40, 24, 37, 49, 29, 47, 57, 34, 68, 49, 44
Offset: 1

Views

Author

Amiram Eldar, Mar 09 2023

Keywords

Comments

Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).

Links

Crossrefs

Cf. A063947, A361316, A361317.
Similar sequences: A001600, A006087.

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}]]; s[1] = 1; s[n_] := n * Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10^4], IntegerQ]
  • 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); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1(ih, ", ")));}

Formula

a(n) = A361316(A063947(n)).
Showing 1-4 of 4 results.

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