A381653 -id:A381653 - cp's OEIS frontend

A382236 Decimal expansion of the multiple prime zeta value primezetamult(2, 2, 2).

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

Offset: 1

Artur Jasinski, Apr 01 2025

Prime multi zeta functions are equivalents of multi zeta functions when successive natural numbers are replaced by successive primes.

For complete list of multiple prime zeta values up to weight 6 see A382234.

			0.003696244163452835378395534632394668...

Cf. A382234, A382235, A381653.

kk = {0, 0}; AppendTo[kk, RealDigits[(PrimeZetaP[2]^3 - 3 PrimeZetaP[2] PrimeZetaP[4] + 2 PrimeZetaP[6])/6, 10, 105][[1]]]; Flatten[kk]

Equals (P(2)^3-3*P(2)*P(4)+2*P(6))/6 where P(n) is Prime zeta function.

A381652 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,1,3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 03 2025

			0.618349560571269307895639260851864...

Index to constants which are multiple zeta values (2,1,3)

Cf. A381651, A381653.

RealDigits[Zeta[3]^2 - 13 Zeta[6]/16, 10, 105][[1]]

PARI
```
zetamult([2,1,3])
```

Equals zeta(3)^2 - 13*zeta(6)/16.

A381654 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3,1) = zetamult(3,1,2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 05 2025

			0.0792213975652071659990328100778...

Cf. A381651, A381651, A381653.

kk = RealDigits[53 Zeta[6]/24 - 3 Zeta[3]^2/2, 10, 105][[1]]; Prepend[kk, 0]

PARI
```
zetamult([3, 1, 2])
```

Equals 53*zeta(6)/24 - 3*zeta(3)^2/2.

A381655 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,1) = zetamult(3, 1, 1, 1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 11 2025

For complete list of formulas of the positive multiple zeta values up to weight 6 see A381651.

			0.04053689727151973782904590793969648...

Cf. A381651, A381652, A381653, A381654, A381656, A381657.

RealDigits[3*Zeta[6]/4 - Zeta[3]^2/2, 10, 106][[1]]

PARI
```
zetamult([3,1,1,1])
```

zetamult(5,1) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^5*n)) = 3*zeta(6)/4 - zeta(3)^2/2 = zetamult(3,1,1,1).

A381656 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2,1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 11 2025

			0.0323090289916698816984064916802...

Cf. A381651, A381652, A381653, A381654, A381655, A381657

RealDigits[-203 Zeta[6]/48 + 3 Zeta[3]^2, 10, 106][[1]]

PARI
```
zetamult([3,2,1])
```

Equals -203*zeta(6)/48 + 3*zeta(3)^2.

A381657 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4, 1, 1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 19 2025

			0.0174898531690114044259344452675...

Artur Jasinski, Complete list of formulas of the positive multiple zeta values up to weight 6 (2025)
Index to constants which are multiple zeta values (4,1,1)

Cf. A381651, A381652, A381653, A381654, A381655, A381656.

RealDigits[23 Zeta[6]/16 - Zeta[3]^2, 10, 106][[1]]

PARI
```
zetamult([4,1,1])
```

Equals 23*A013664/16 - A002117^2.

cp's OEIS Frontend

A382236 Decimal expansion of the multiple prime zeta value primezetamult(2, 2, 2).

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A381652 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,1,3).

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A381654 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3,1) = zetamult(3,1,2).

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A381655 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,1) = zetamult(3, 1, 1, 1).

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A381656 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2,1).

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A381657 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4, 1, 1).

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