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.

A358659 Decimal expansion of the asymptotic mean of the ratio between the number of exponential unitary divisors and the number of exponential divisors.

Original entry on oeis.org

9, 8, 4, 8, 8, 3, 6, 4, 1, 8, 7, 7, 2, 2, 8, 2, 9, 4, 0, 9, 5, 3, 7, 0, 1, 3, 8, 0, 4, 8, 9, 6, 1, 1, 3, 7, 6, 4, 7, 3, 1, 6, 3, 2, 2, 2, 2, 7, 0, 5, 8, 1, 3, 4, 5, 5, 0, 0, 6, 3, 6, 2, 3, 5, 5, 0, 2, 2, 3, 9, 6, 8, 0, 6, 5, 9, 0, 8, 2, 3, 8, 0, 0, 8, 1, 8, 9, 3, 8, 0, 9, 5, 5, 7, 4, 0, 8, 7, 6, 9, 1, 3, 3, 4, 4
Offset: 0

Views

Author

Amiram Eldar, Nov 25 2022

Keywords

Examples

			0.984883641877228294095370138048961137647316322227058...

Links

  • Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.

Crossrefs

Cf. A049419, A278908.
Similar sequences: A307869, A308042, A308043.

Programs

  • Mathematica
    r[n_] := 2^PrimeNu[n]/DivisorSigma[0, n]; $MaxExtraPrecision = 500; m = 500; f[x_] := Log[1 + Sum[x^e*(r[e] - r[e - 1]), {e, 4, m}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[f[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

Formula

Equals lim_{m->oo} (1/m) Sum_{k=1..m} A278908(k)/A049419(k).
Equals Product_{p prime} (1 + Sum_{e >= 4} (r(e) - r(e-1))/p^e), where r(e) = A278908(e)/A049419(e).

A366586 Decimal expansion of the asymptotic mean of the ratio between the number of cubefree divisors and the number of squarefree divisors.

Original entry on oeis.org

1, 2, 4, 2, 5, 3, 4, 1, 8, 6, 2, 2, 4, 6, 7, 7, 2, 8, 6, 9, 5, 9, 6, 3, 0, 0, 0, 6, 2, 9, 4, 3, 3, 7, 7, 0, 8, 0, 0, 0, 1, 5, 2, 5, 3, 3, 0, 5, 8, 9, 0, 5, 9, 8, 0, 1, 9, 8, 3, 2, 2, 6, 8, 4, 7, 1, 5, 9, 2, 4, 7, 4, 4, 9, 2, 0, 0, 5, 9, 2, 9, 5, 1, 5, 5, 5, 2, 8, 3, 3, 0, 5, 8, 6, 2, 6, 6, 4, 9, 1, 9, 2, 9, 0, 6
Offset: 1

Views

Author

Amiram Eldar, Oct 14 2023

Keywords

Comments

For a positive integer k the ratio between the number of cubefree divisors and the number of squarefree divisors is r(k) = A073184(k)/A034444(k).
r(k) >= 1 with equality if and only if k is squarefree (A005117).
The indices of records of this ratio are the squares of primorial numbers (A061742), and the corresponding record values are r(A061742(k)) = (3/2)^k. Therefore, this ratio is unbounded.
The asymptotic second raw moment is = Product_{p prime} (1 + 5/(4*p^2)) = 1.67242666864454336962... and the asymptotic standard deviation is 0.35851843008068965078... .

Examples

			1.24253418622467728695963000629433770800015253305890...

Crossrefs

Cf. A005117, A034444, A061742, A073184.
Similar constants: A307869, A308042, A308043, A358659, A361059, A361060, A361061, A361062, A366587 (mean of the inverse ratio).

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{0, -(1/2)}, {0, 1}, m]; RealDigits[Exp[NSum[Indexed[c, n] * PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]
  • PARI
    prodeulerrat(1 + 1/(2*p^2))

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A073184(k)/A034444(k).
Equals Product_{p prime} (1 + 1/(2*p^2)).
In general, the asymptotic mean of the ratio between the number of (k+1)-free divisors and the number of k-free divisors, for k >= 2, is Product_{p prime} (1 + 1/(k*p^2)).

A366587 Decimal expansion of the asymptotic mean of the ratio between the number of squarefree divisors and the number of cubefree divisors.

Original entry on oeis.org

8, 5, 6, 2, 0, 0, 5, 0, 7, 9, 3, 7, 4, 7, 7, 1, 4, 9, 3, 9, 7, 2, 8, 1, 0, 8, 9, 5, 9, 5, 1, 6, 0, 4, 0, 4, 9, 8, 8, 4, 9, 0, 3, 1, 5, 8, 4, 1, 3, 2, 7, 1, 3, 1, 8, 5, 9, 6, 9, 5, 5, 8, 0, 3, 4, 0, 3, 8, 6, 6, 0, 8, 9, 6, 0, 1, 1, 9, 5, 9, 2, 1, 0, 5, 5, 5, 3, 0, 9, 0, 7, 8, 0, 9, 2, 3, 1, 4, 3, 4, 9, 2, 7, 3, 9
Offset: 0

Views

Author

Amiram Eldar, Oct 14 2023

Keywords

Comments

For a positive integer k the ratio between the number of squarefree divisors and the number of cubefree divisors is r(k) = A034444(k)/A073184(k).
r(k) <= 1 with equality if and only if k is squarefree (A005117).
The asymptotic second raw moment is = Product_{p prime} (1 - 5/(9*p^2)) = 0.76780883634140395932... and the asymptotic standard deviation is 0.29730736888962774256... .

Examples

			0.85620050793747714939728108959516040498849031584132...

Crossrefs

Cf. A005117, A034444, A073184.
Similar constants: A307869, A308042, A308043, A358659, A361059, A361060, A361061, A361062, A366586 (mean of the inverse ratio).

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{0, 1/3}, {0, -(2/3)}, m]; RealDigits[Exp[NSum[Indexed[c, n] * PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]
  • PARI
    prodeulerrat(1 - 1/(3*p^2))

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A034444(k)/A073184(k).
Equals Product_{p prime} (1 - 1/(3*p^2)).
In general, the asymptotic mean of the ratio between the number of k-free divisors and the number of (k-1)-free divisors, for k >= 3, is Product_{p prime} (1 - 1/(k*p^2)).
Showing 1-3 of 3 results.

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