A334585 -id:A334585 - cp's OEIS frontend

A120296 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^4.

1, 15, 1231, 19615, 12280111, 4090037, 9824498837, 157151464517, 38193952437631, 7637983935923, 111835788321880643, 111830093529238643, 3194097388508809394723, 3194009594644356242723, 15970381078317764649391

Offset: 1

Alexander Adamchuk, Jul 10 2006

p divides a(p-1) for prime p > 2 - similar to Wolstenholme's theorem for A007406(n) (= numerator of Sum_{k=1..n} 1/k^2) and for A007410(n) (= numerator of Sum_{k=1..n} 1/k^4).

Lim_{n -> infinity} a(n)/A334585(n) = A267315 = (7/8)*A013662. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 15/16, 1231/1296, 19615/20736, 12280111/12960000, 4090037/4320000, 9824498837/10372320000, ... = A120296/A334585. - _Petros Hadjicostas_, May 06 2020

Cf. A007406, A007410, A013662, A119682, A267315, A334585 (denominators).

Numerator[Table[Sum[(-1)^(k+1)/k^4,{k,1,n}],{n,1,20}]]

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^4)); \\ Michel Marcus, May 07 2020

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/k^4).

Name edited by Petros Hadjicostas, May 07 2020

A267315 Decimal expansion of the Dirichlet eta function at 4.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

			eta(4) = 1/1^4 - 1/2^4 + 1/3^4 - 1/4^4 + 1/5^4 - 1/6^4 + ... = 0.9470328294972459175765032344735219149279070829288860...

OEIS Wiki, Euler's alternating zeta function.
Eric Weisstein's World of Mathematics, Dirichlet Eta Function.
Wikipedia, Dirichlet Eta Function.

Cf. A002162, A013662, A072691, A120296, A197070, A334585.

pi:= 7*Pi(RealField(110))^4 / 720; Reverse(Intseq(Floor(10^100*pi))); // Vincenzo Librandi, Feb 04 2016

Mathematica
```
RealDigits[(7 Pi^4)/720, 10, 120][[1]]
```

7*Pi^4/720 \\ Michel Marcus, Feb 01 2016

s = RLF(0); s
RealField(110)(s)
for i in range(1,10000): s += -((-1)^i/((i)^4))
print(s) # Terry D. Grant, Aug 04 2016

eta(4) = Sum_{k > 0} (-1)^(k+1)/k^4 = (7*Pi^4)/720.

eta(4) = Lim_{n -> infinity} A120296(n)/A334585(n) = (7/8)*A013662. - Petros Hadjicostas, May 07 2020

cp's OEIS Frontend

A120296 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^4.

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