A175639 -id:A175639 - cp's OEIS frontend

A330523 Decimal expansion of Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4).

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

Offset: 0

Vaclav Kotesovec, Dec 17 2019

			0.5358961538283379998085026313185459506482223745141452711510108346133288119...

Cf. A055653, A062354, A065465, A175639, A256392, A332713.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 + p^3 - p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 50]], {t, 10, 100, 10}]

(6/Pi^2) * prodeulerrat(1 - 1/(p^2*(p+1))) \\ Amiram Eldar, Sep 08 2020

Equals (6/Pi^2) * A065465. - Amiram Eldar, Sep 08 2020

A183030 Decimal expansion of Sum_{j>=1} tau(j)/j^3 = zeta(3)^2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Dec 18 2010

This is the zeta-function Sum_{j>=1} A000005(j)/j^s evaluated at s=3. At s=2 we find A098198, at s=4 A183031.

Since tau(n)/n^3 is a multiplicative function, one finds an Euler product for the sum, which is expanded with an Euler transformation to a product of Riemann zeta functions as in A175639 for numerical evaluation.

			1.4449407984336342339136.. = 1 +2/2^3 +2/3^3 +3/4^3 +2/5^3 +4/6^3 +2/7^3+...

Cf. A000005, A002117, A098198, A175639, A183031.

Maple
```
evalf(Zeta(3)^2);
```

RealDigits[Zeta[3]^2, 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

zeta(3)^2 \\ Charles R Greathouse IV, Mar 04 2015

Equals the Euler product Product_{p= A000040} (1+ (2*p^s-1)/(p^s-1)^2) at s=3, or the square of A002117.

A183031 Decimal expansion of Sum_{j>=1} tau(j)/j^4 = Pi^8/8100.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Dec 18 2010

This is the zeta-function Sum_{j>=1} A000005(j)/j^s evaluated at s=4. At s=2, we find A098198; at s=3, A183030.

Since tau(n)/n^4 is a multiplicative function, one finds an Euler product for the sum, which is expanded with an Euler transformation to a product of Riemann zeta functions as in A175639 for numerical evaluation.

			1.1714235822309350626084... = 1 + 2/2^4 + 2/3^4 + 3/4^4 + 2/5^4 + 4/6^4 + 2/7^4 + ...

Index entries for transcendental numbers

Cf. A000005, A098198, A183030, A175639.

Maple
```
evalf(Pi^8/8100) ;
```

RealDigits[Zeta[4]^2, 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

zeta(4)^2 \\ Charles R Greathouse IV, Mar 04 2015

Equals the Euler product Product_{p prime} (1 + (2*p^s - 1)/(p^s - 1)^2) at s=4, which is the square of A013662.

cp's OEIS Frontend

A330523 Decimal expansion of Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4).

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A183030 Decimal expansion of Sum_{j>=1} tau(j)/j^3 = zeta(3)^2.

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A183031 Decimal expansion of Sum_{j>=1} tau(j)/j^4 = Pi^8/8100.

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Maple

Mathematica

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