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 1-10 of 10 results.

A307870 Numbers k with record values of the ratio d(k)/ud(k) between the number of divisors and the number of unitary divisors.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 256, 432, 576, 864, 1296, 1728, 2592, 3456, 5184, 6912, 10368, 15552, 20736, 31104, 41472, 62208, 82944, 93312, 124416, 186624, 248832, 373248, 497664, 746496, 995328, 1119744, 1492992, 2239488, 2592000, 2985984, 3888000, 5184000, 7776000
Offset: 1

Views

Author

Amiram Eldar, May 02 2019

Keywords

Comments

Numbers k with d(k)/2^omega(k) > d(j)/2^omega(j) for all j < k, where d(k) is the number of divisors of k (A000005), and omega(k) is the number of distinct prime factors of k (A001221), so 2^omega(k) is the number of unitary divisors of k (A034444).
Subsequence of A025487.
The first term that is divisible by the k-th prime is 4, 432, 2592000, 53343360000, 134190022982400000, 35377857659079936000000, 160601747163451186424832000000, 35800939973308629849857487360000000, ...
All the terms are powerful (A001694), since if p is a prime factor of k with multuplicity 1, then k and k/p have the same ratio.

Examples

			All squarefree numbers k have d(k)/ud(k) = 1. Thus 4, the first nonsquarefree number, has a record value of d(4)/ud(4) = 3/2 and thus it is in the sequence.

Links

Crossrefs

Cf. A000005, A001221, A001694, A025487, A034444, A285906, A307869.

Programs

  • Mathematica
    r[n_] := DivisorSigma[0, n]/(2^PrimeNu[n]); rm = 0; n = 1; s = {}; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[s, n]]; n++, {10^7}]; s

A361062 Decimal expansion of the asymptotic mean of A073184(k)/A000005(k), the ratio between the number of cubefree divisors and the number of divisors.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Mar 01 2023

Keywords

Examples

			0.939974352176477078470442562386025726769842310977996...

Crossrefs

Cf. A000005, A073184, A361061 (mean of the inverse ratio).
Cf. A307869, A308043 (squarefree analog).

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; f[p_] := -(p-1)*(1+4*p+6*p^2*Log[1-1/p])/(2*p^2); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n]), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A073184(k)/A000005(k).
Equals Product_{p prime} (-(p-1)*(1+4*p+6*p^2*log(1-1/p))/(2*p^2)).

A361059 Decimal expansion of the asymptotic mean of A000005(k)/A286324(k), the ratio between the number of divisors and the number of bi-unitary divisors.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Mar 01 2023

Keywords

Examples

			1.158854572650312100164480196393175149039104318857395...

Links

Crossrefs

Cf. A000005, A286324, A361060 (mean of the inverse ratio).
Cf. A307869 (unitary analog), A308043.

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 - (p - 1)*Log[1 - 1/p^2]/(2*p); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n], {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 106][[1]]

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A000005(k)/A286324(k).
Equals Product_{p prime} (1 - (p-1)*log(1 - 1/p^2)/(2*p)).

A361060 Decimal expansion of the asymptotic mean of A286324(k)/A000005(k), the ratio between the number of bi-unitary divisors and the number of divisors.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Mar 01 2023

Keywords

Examples

			0.901241806826482255139197485094387558982811533821787...

Links

Crossrefs

Cf. A000005, A286324, A361059 (mean of the inverse ratio).
Cf. A307869, A308043 (unitary analog).

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; f[p_] := 2 - 1/p - (p - 1)*Log[(p + 1)/(p - 1)]/2; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n], {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A286324(k)/A000005(k).
Equals Product_{p prime} (2 - 1/p - (p-1)*log((p+1)/(p-1))/2).

A361061 Decimal expansion of the asymptotic mean of A000005(k)/A073184(k), the ratio between the number of divisors and the number of cubefree divisors.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Mar 01 2023

Keywords

Examples

			1.109049677998737336345288587781671766009752629677303...

Crossrefs

Cf. A000005, A073184, A361062 (mean of the inverse ratio).
Cf. A307869 (squarefree analog), A308043.

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 + 1/(3*(p - 1)*p^2); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n]), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
  • PARI
    prodeulerrat(1 + 1/(3*(p-1)*p^2))

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A000005(k)/A073184(k).
Equals Product_{p prime} (1 + 1/(3*(p-1)*p^2)).

A308042 Decimal expansion of the asymptotic mean of d_3(k)/ud_3(k), where d_3(k) is the number of ordered factorizations of k as product of 3 divisors (A007425) and ud_3(k) = 3^omega(k) is the unitary analog of d_3 (A074816).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, May 10 2019

Keywords

Examples

			2.22416248380186958442174889454690037857600080851428...

Links

Crossrefs

Cf. A001221, A007425, A074816, A307869.

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{6, -16, 64/3, -32/3}, {0, 8, 32, 224/3}, m]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n/2^n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
  • PARI
    prodeulerrat((1 - 1/p) * (2 + (1 - 1/p)^(-3))/3) \\ Amiram Eldar, Sep 16 2024

Formula

Equals Product_{p prime} ((1 - 1/p) * (2 + (1 - 1/p)^(-3))/3).

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)).

A380602 Decimal expansion of the asymptotic mean of the ratio A000005(k)/A322483(k).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jan 27 2025

Keywords

Examples

			1.23587977752617354871093805318945110447752750370305...

Crossrefs

Cf. A000005, A322483, A380601 (mean of the inverse ratio).
Similar constants: A307869, A361059, A361061.

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; m = 1000; f[x_] := 2/x - (1 + 1/x - 2/x^2)*Log[1-x^2]/x; c = Rest[CoefficientList[Series[Log[f[x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]
  • PARI
    default(realprecision, 120); default(parisize, 30000000);
my(m = 1024, x = 'x + O('x^m), v); v = Vec(2/x - (1 + 1/x - 2/x^2)*log(1-x^2)/x); prodeulerrat(sum(i=1, #v, v[i]/p^(i-1)))

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A000005(k)/A322483(k).
Equals Product_{p prime} (2*p - (1+p-2*p^2)*log(1-1/p^2)*p).
Showing 1-10 of 10 results.

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