A226317 -id:A226317 - cp's OEIS frontend

A371331 Decimal expansion of Sum_{k>=1} 1/(k^(1/3)*(1+k)).

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

Offset: 1

R. J. Mathar, Mar 19 2024

			2.790018293097153329933562762205...

Cf. A226317 (for k^1/2), A371332 (k^1/4), A371333 (k^1/5).

sumalt(i=0,(-1)^i*zeta(4/3+i)) \\ Hugo Pfoertner, Mar 19 2024

Equals Sum_{i>=0} (-1)^i*zeta(4/3+i).

A371332 Decimal expansion of Sum_{k>=1} 1/(k^(1/4)*(1+k)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Mar 19 2024

			3.7478347438448875778880546...

Cf. A226317 (for k^1/2), A371331 (k^1/3), A371333 (k^1/5).

sumalt(i=0,(-1)^i*zeta(5/4+i)) \\ Hugo Pfoertner, Mar 19 2024

Equals Sum_{i>=0} (-1)^i*zeta(5/4+i).

A371333 Decimal expansion of Sum_{k>=1} 1/(k^(1/5)*(1+k)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Mar 19 2024

			4.71961753816261448360578272400...

Cf. A226317 (for k^1/2), A371331 (k^1/3), A371332 (k^1/4).

sumalt(i=0,(-1)^i*zeta(6/5+i)) \\ Hugo Pfoertner, Mar 19 2024

Equals Sum_{i>=0} (-1)^i*zeta(6/5+i).

A234014 Decimal expansion of Sum_{x>=2} 1/((x - 1) sqrt(x)) = Sum_{k>=1} (zeta(k+1/2) - 1).

Original entry on oeis.org

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

Offset: 1

David Brink and Robert G. Wilson v, Dec 18 2013

			2.184009470267851952894734157852949070443908406263229420200251207928354...

David Brink, The Spiral of Theodorus and Sums of Zeta-values at the Half-Integers, The American Mathematical Monthly, Vol 119, No. 9 (November 2012), page 785.

Cf. A105459, A226317, A234015.

RealDigits[ Sum[ Zeta[k + 1/2] - 1, {k, 1, 355}], 10, 111][[1]]

PARI
```
sum(k=1,340,zeta(k+1/2)-1)
```

A234015 K' (K being A105459) = Sum_{k>=0} (Zeta(k+1/2)-1)/(2k+1), negated.

Original entry on oeis.org

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

Offset: 1

David Brink and Robert G. Wilson v, Dec 18 2013

			1.8265078108584785588157654061303219730994914849434906683229013637764992...

David Brink, The Spiral of Theodorus and Sums of Zeta-values at the Half-Integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), page 785.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 663 (constant K').

Cf. A105459, A226317, A234014.

RealDigits[ Sum[ (Zeta[k + 1/2] - 1)/(2 k + 1), {k, 0, 370}], 10,  111][[1]]

PARI
```
sum(k=0,340,(zeta(k+1/2)-1)/(2*k+1))
```

cp's OEIS Frontend

A371331 Decimal expansion of Sum_{k>=1} 1/(k^(1/3)*(1+k)).

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A371332 Decimal expansion of Sum_{k>=1} 1/(k^(1/4)*(1+k)).

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A371333 Decimal expansion of Sum_{k>=1} 1/(k^(1/5)*(1+k)).

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A234014 Decimal expansion of Sum_{x>=2} 1/((x - 1) sqrt(x)) = Sum_{k>=1} (zeta(k+1/2) - 1).

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A234015 K' (K being A105459) = Sum_{k>=0} (Zeta(k+1/2)-1)/(2k+1), negated.

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