A157289 -id:A157289 - cp's OEIS frontend

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.

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)).

A157293 Decimal expansion of zeta(3)/zeta(9).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 26 2009

The product Product_{p = primes = A000040} (1+1/p^3+1/p^6). The product over (1+2/p^3+1/p^6) equals A157289^2.

			1.19964753964713... = (1+1/2^3+1/2^6)*(1+1/3^3+1/3^6)*(1+1/5^3+1/5^6)*(1+1/7^3+1/7^6)*...

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, eq. (23).

Cf. A002117, A013667, A030078, A030516, A157289.

Maple
```
evalf(Zeta(3)/Zeta(9)) ;
```

RealDigits[Zeta[3]/Zeta[9], 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Equals A002117/A013667 = Product_{i>=1} (1+1/A030078(i)+1/A030516(i)) .

A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e(e+1)/2) + e) p^(e-1) for prime p and e > 0.

Original entry on oeis.org

1, 3, 5, 12, 9, 15, 13, 40, 30, 27, 21, 60, 25, 39, 45, 120, 33, 90, 37, 108, 65, 63, 45, 200, 90, 75, 153, 156, 57, 135, 61, 336, 105, 99, 117, 360, 73, 111, 125, 360, 81, 195, 85, 252, 270, 135, 93, 600, 182, 270, 165, 300, 105, 459, 189, 520, 185, 171, 117, 540, 121, 183, 390, 896

Offset: 1

Werner Schulte, Nov 09 2022

Cf. A000010, A005361, A018804, A112526.

Cf. A001620, A002117, A013664, A244115, A157289, A306016.

f[p_, e_] := ((p - 1)*(1 + e*(e + 1)/2) + e)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2022 *)

a(n) = { my (f=factor(n), p, e, v=1); for (k=1, #f~, p=f[k,1]; e=f[k,2]; v *= ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1)); return (v) } \\ Rémy Sigrist, Jan 18 2023

a(n) = Sum_{k=1..n} gcd(k, n) * A005361(gcd(k, n)) for n > 0.
Equals Dirichlet convolution of A000010 and n * A005361.
Dirichlet g.f.: (zeta(s-1)^2 * zeta(2*s-2) * zeta(3*s-3)) / (zeta(s) * zeta(6*s-6)).
Equals Dirichlet convolution of A018804 and A112526.
Sum_{k=1..n} a(k) ~ (zeta(3)/(2*zeta(6))) * n^2 * (log(n) + 2*gamma - 1/2 + zeta'(2)/zeta(2) + 3*zeta'(3)/zeta(3) + 6*zeta'(6)/zeta(6)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 13 2024

A373702 Decimal expansion of (2 - zeta(2))zeta(2)zeta(3)/zeta(6).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jun 13 2024

			0.69010488251022497818773002567827532644066623131...

Steve Fan, The shifted prime-divisor function over shifted primes, arXiv:2406.05217 [math.NT], 2024. See p. 34.

Cf. A002117, A013661, A013664, A082695, A152416, A157289, A157292, A183699.

RealDigits[(2-Zeta[2])Zeta[2]Zeta[3]/Zeta[6],10,100][[1]]

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A157293 Decimal expansion of zeta(3)/zeta(9).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1) for prime p and e > 0.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A373702 Decimal expansion of (2 - zeta(2))*zeta(2)*zeta(3)/zeta(6).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e(e+1)/2) + e) p^(e-1) for prime p and e > 0.

A373702 Decimal expansion of (2 - zeta(2))zeta(2)zeta(3)/zeta(6).