A085361 -id:A085361 - cp's OEIS frontend

A244625 Decimal expansion of Product_{n>1} (1 - 1/n^2)^(1/n).

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

Offset: 0

Jean-François Alcover, Jul 02 2014

			0.7993704013063328789872528539753525668777...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.9 Alladi-Grinstead Constant, p. 122.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A085361, A242623, A242624.

SetDefaultRealField(RealField(100)); L:=RiemannZeta();  Exp(-(&+[(Evaluate(L, 2*n+1)-1)/n: n in [1..10^3]])); // G. C. Greubel, Nov 15 2018

evalf(exp(-sum((Zeta(2*n+1)-1)/n, n=1..infinity)), 120); # Vaclav Kotesovec, Dec 11 2015

digits = 102; Exp[-NSum[(Zeta[2*n+1]-1)/n, {n, 1, Infinity}, NSumTerms -> 300, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First

default(realprecision, 100); exp(-suminf(n=1, (zeta(2*n+1)-1)/n)) \\ G. C. Greubel, Nov 15 2018

numerical_approx(exp(-sum((zeta(2*n+1)-1)/n for n in [1..1000])), digits=100) # G. C. Greubel, Nov 15 2018

Equals exp(-Sum_{n>0} (zeta(2*n+1) - 1)/n).
Equals A242623 * A242624.
Also equals A242623 * exp(-A085361).

A345384 Decimal expansion of Sum_{k>=2} (-1)^k * zeta''(k).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 17 2021

			1.797744327618368050843854084033051615798915710844403144091653059609067...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 528.

Cf. A085361, A345385.

evalf(Sum((-1)^k*Zeta(2, k), k = 2..infinity), 120);

RealDigits[Sum[(-1)^k*Zeta''[k], {k, 2, 1000}], 10, 110][[1]]

PARI
```
sumalt(k=2, (-1)^k * zeta''(k))
```

Equals Sum_{k>=1} log(k)^2 / (k*(k+1)).

A345385 Decimal expansion of Sum_{k>=2} zeta''(k).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 17 2021

			2.3363131760589332919965154813777122447557697943302707967767827087829289...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
Michael I. Shamos, A catalog of the real numbers 2011, p. 355.

Cf. A085361, A131688, A345384.

evalf(Sum(Zeta(2, k), k = 2..infinity), 120);

RealDigits[Sum[Zeta''[k], {k, 2, 1000}], 10, 110][[1]]

PARI
```
suminf(k=2, zeta''(k))
```

Equals Sum_{k>=1} log(k+1)^2 / (k*(k+1)).

cp's OEIS Frontend

A244625 Decimal expansion of Product_{n>1} (1 - 1/n^2)^(1/n).

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A345384 Decimal expansion of Sum_{k>=2} (-1)^k * zeta''(k).

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A345385 Decimal expansion of Sum_{k>=2} zeta''(k).

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