A361062 -id:A361062 - cp's OEIS frontend

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.

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

Amiram Eldar, Mar 01 2023

			1.109049677998737336345288587781671766009752629677303...

Cf. A000005, A073184, A361062 (mean of the inverse ratio).

Cf. A307869 (squarefree analog), A308043.

$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))
```

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)).

A383157 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 7, 1, 3, 3, 2, 1, 7, 1, 7, 3, 3, 1, 13, 1, 3, 3, 7, 1, 7, 1, 5, 3, 3, 3, 13, 1, 3, 3, 13, 1, 7, 1, 7, 7, 3, 1, 21, 1, 7, 3, 7, 1, 13, 3, 13, 3, 3, 1, 5, 1, 3, 7, 3, 3, 7, 1, 7, 3, 7, 1, 11, 1, 3, 7, 7, 3, 7, 1, 21, 2, 3, 1, 5, 3

Offset: 1

Amiram Eldar, Apr 18 2025

a(n) depends only on the prime signature of n (A118914).

			Fractions begin with 0, 1/2, 1/2, 1, 1/2, 3/4, 1/2, 3/2, 1, 3/4, 1/2, 7/6, ...
4 has 3 divisors: 1, 2 = 2^1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0, 1 and 2, respectively. Therefore, a(4) = numerator((0 + 1 + 2)/3) = numerator(1) = 1.
12 has 6 divisors: 1, 2 = 2^1, 3 = 3^1, 4 = 2^2, 6 = 2 * 3 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 1, 2, 1 and 2, respectively. Therefore, a(12) = numerator((0 + 1 + 1 + 2 + 1 + 2)/6) = numerator(7/6) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of (Sum_{k=1..n} a(k)/A383158(k))/(c_1*n - c_2*n/sqrt(log(n))) for n = 10^(1..10).

Cf. A000005, A001248, A051903, A118914, A308043, A345231, A361062, A383156, A383158 (denominators).

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := Numerator[DivisorSum[n, emax[#] &] / DivisorSigma[0, n]]; Array[a, 100]

emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = my(f = factor(n)); numerator(sumdiv(n, d, emax(d)) / numdiv(f));

a(n) = numerator(Sum_{d|n} A051903(d) / A000005(n)) = numerator(A383156(n) / A000005(n)).
a(n)/A383158(n) = 1 if and only if n is a square of a prime (A001248).
Sum_{k=1..n} a(k)/A383158(k) ~ c_1 * n - c_2 * n /sqrt(log(n)), where c_1 = m(2) + Sum_{k>=3} (k-1) * (m(k) - m(k-1)) = 1.27968644485944694957... is the asymptotic mean of the fractions a(k)/A383158(k), m(k) = Product_{p prime} (1 + (1-1/p) * Sum_{i>=k} (k/(i+1) - 1)/p^i is the asymptotic mean of the ratio between the number of k-free divisors and the number of divisors, e.g., m(2) = A308043 and m(3) = A361062, and c_2 = A345231 = 0.54685595528047446684... .

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

Amiram Eldar, Oct 14 2023

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... .

			1.24253418622467728695963000629433770800015253305890...

Cf. A005117, A034444, A061742, A073184.

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

$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))
```

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)).

A383058 Decimal expansion of the asymptotic mean of A365498(k)/A034444(k), the ratio between the number of cubefree unitary divisors and the number of unitary divisors over the positive integers.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Apr 15 2025

The asymptotic mean of the inverse ratio A034444(k)/A365498(k) is zeta(3)/zeta(6) (A157289).

In general, the asymptotic mean of the inverse ratio, between the number of unitary divisors and the number of k-free (i.e., not divisible by a k-th power other than 1) unitary divisors over the positive integers, for k >= 2, is zeta(k)/zeta(2*k).

			0.91429441180198062448296176452156718437854669178193...

The unitary analog of A361062.

Cf. A034444, A157289, A365498, A383057.

$MaxExtraPrecision = 300; m = 300; f[p_] := 1 - 1/(2*p^3); 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, 120][[1]]

PARI
```
prodeulerrat(1 - 1/(2*p^3))
```

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A365498(k)/A034444(k).
Equals Product_{p prime} (1 - 1/(2*p^3)).
In general, the asymptotic mean of the ratio between the number of k-free unitary divisors and the number of unitary divisors over the positive integers, for k >= 2, is Product_{p prime} (1 - 1/(2*p^k)).

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

Amiram Eldar, Oct 14 2023

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... .

			0.85620050793747714939728108959516040498849031584132...

Cf. A005117, A034444, A073184.

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

$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))
```

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)).

A380601 Decimal expansion of the asymptotic mean of the ratio A322483(k)/A000005(k).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jan 27 2025

			0.85980677933034363311244767594941832466515809551385...

Cf. A000005, A322483, A380602 (mean of the inverse ratio).

Similar constants: A308043, A361060, A361062.

$MaxExtraPrecision = 1000; m = 1000; f[x_] := (2*x + 3*(x-1)*Log[1 - x] + (x-1)*Log[1+x])/(4*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]]

default(realprecision, 120); default(parisize, 30000000);
my(m = 1024, x = 'x + O('x^m), v); v = Vec((2*x + 3*(x-1)*log(1-x) + (x-1)*log(1+x))/(4*x)); prodeulerrat(sum(i=1, #v, v[i]/p^(i-1)))

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A322483(k)/A000005(k).

Equals Product_{p prime} (1/2 - ((p-1)/4) * (3*log(1-1/p) + log(1+1/p))).

cp's OEIS Frontend

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.

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A383157 a(n) is the numerator of the mean of the maximum exponents in the prime factorizations of the divisors of n.

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

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A383058 Decimal expansion of the asymptotic mean of A365498(k)/A034444(k), the ratio between the number of cubefree unitary divisors and the number of unitary divisors over the positive integers.

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

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Programs

Mathematica

PARI

Formula

A380601 Decimal expansion of the asymptotic mean of the ratio A322483(k)/A000005(k).

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Formula