A366586 -id:A366586 - cp's OEIS frontend

A374785 Numbers whose unitary divisors have a mean unitary abundancy index that is larger than 2.

223092870, 281291010, 300690390, 6469693230, 6915878970, 8254436190, 8720021310, 9146807670, 9592993410, 10407767370, 10485364890, 10555815270, 11125544430, 11532931410, 11797675890, 11823922110, 12095513430, 12328305990, 12598876290, 12929686770, 13162479330

Offset: 1

Amiram Eldar, Jul 20 2024

nonn

Numbers k such that A374783(k)/A374784(k) > 2.
The least odd term is A070826(43) = 5.154... * 10^74, and the least term that is coprime to 6 is Product_{k=3..219} prime(k) = 1.0459... * 10^571.
The least nonsquarefree (A013929) term is a(613) = 148802944290 = 2 * 3 * 5 * 7 * 11 * 13 * 17 *19 * 23^2 * 29.
All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, A374783/(k)/A374784(k) < Product_{p prime} (1 + 1/(2*p)) = 1.242534... (A366586).

			223092870 is a term since A374783(223092870)/A374784(223092870) = 666225/330752 = 2.014... > 2.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A052485.
Cf. A001221, A001694, A013929, A034683, A070826, A366586, A374783, A374784.
Similar sequences: A245214, A374788.

f[p_, e_] := 1 + 1/(2*p^e); r[1] = 1; r[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[4*10^8], s[#] > 2 &]

is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + 1/(2*f[i,1]^f[i,2])) > 2;}

A001221(a(n)) >= 9.

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)).

cp's OEIS Frontend

A374785 Numbers whose unitary divisors have a mean unitary abundancy index that is larger than 2.

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