A076813 -id:A076813 - cp's OEIS frontend

A093720 Decimal expansion of Sum_{n >= 2} zeta(n)/n!.

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

Offset: 1

Eric W. Weisstein, Apr 12 2004

			1.078188729575818482758265436769832381707219...

G. C. Greubel, Table of n, a(n) for n = 1..10000
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Cf. A076813, A093721, A269574, A269611, A269720, A269768.

evalf(Sum(exp(1/n)-1-1/n, n=1..infinity), 120); # Vaclav Kotesovec, Mar 04 2016

digits = 99; ClearAll[z, rd]; z[k_] := z[k] = z[k-1] + N[Sum[Zeta[n]/n!, {n, 2^(k-1) + 1, 2^k}], digits]; z[0] = 0; rd[k_] := rd[k] = RealDigits[z[k]][[1]]; rd[0]; rd[k = 1]; While[ rd[k] != rd[k-1], k++]; rd[k] (* Jean-François Alcover, Nov 09 2012 *)

suminf(n=2, zeta(n)/n!) \\ Michel Marcus, Mar 15 2017

Equals Sum_{k>=1} (exp(1/k) - 1 - 1/k). - Vaclav Kotesovec, Mar 04 2016

Equals Integral_{x=0..oo} exp(1/(x^2 + 1))*sin(x/(x^2 + 1))*(coth(Pi*x) - 1) dx + A091725 - 2*A001620 - exp(1)/2 + 3/2. - Velin Yanev, Nov 14 2024

Corrected by Fredrik Johansson, Mar 19 2006

A080729 Decimal expansion of the infinite product of zeta functions for even arguments.

Original entry on oeis.org

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

Offset: 1

Deepak R. N (deepak_rn(AT)safe-mail.net), Mar 08 2003

By elementary estimates, the constant lies in the open interval (Pi/6, exp(3/4)). - Bernd C. Kellner, May 18 2024

			1.82101745149929239040672513222600684857...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 658.
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers, Vol. 9 (2009), Article #A08, pp. 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
Eric Weisstein's World of Mathematics, Abelian group.

Cf. A000688, A021002, A076813, A080730, A369634.

RealDigits[Product[Zeta[2n],{n,500}],10,110][[1]] (* Harvey P. Dale, Jan 31 2012 *)

prodinf(k=1, zeta(2*k)) \\ Vaclav Kotesovec, Jan 29 2024

Decimal expansion of zeta(2)*zeta(4)*...*zeta(2k)*...
If u(k) denotes the number of Abelian groups with group order k (A000688), then Product_{k>=1} zeta(2*k) = Sum_{k>=1} u(k)/k^2. - Benoit Cloitre, Jun 25 2003
Equals A021002/A080730. - Amiram Eldar, Jan 31 2024
This constant C is connected with the product of values of the Dedekind eta function on the upper imaginary axis. The product runs over the primes, where i is the imaginary unit: 1/C = Product_{prime p} (p^(1/12) * eta(i * log(p) / Pi)). - Bernd C. Kellner, May 18 2024

More terms from Benoit Cloitre, Mar 08 2003

A093721 Decimal expansion of Sum_{n>=1} zeta(2n)/(2n)!.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 12 2004

			0.86900199196290899881105480556139568889249484188057785071064577856...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Cf. A076813, A093720, A269574, A269611.

evalf(Sum(cosh(1/n)-1, n=1..infinity), 120); # Vaclav Kotesovec, Mar 04 2016

digits = 105; z[k_] := z[k] = z[k-1] + N[Sum[Zeta[2n]/(2n)!, {n, 2^(k-1) + 1, 2^k}], digits]; z[0] = N[Pi^2/12, digits]; rd[k_] := rd[k] = RealDigits[z[k]][[1]]; rd[0]; rd[k = 1]; While[rd[k] != rd[k-1], k++]; rd[k] (* Jean-François Alcover, Nov 09 2012 *)

suminf(n=1, zeta(2*n)/(2*n)!) \\ Michel Marcus, Mar 20 2017

Equals Sum_{k>=1} (cosh(1/k) - 1). - Vaclav Kotesovec, Mar 04 2016

A084862 Continued fraction expansion of Sum_{n>=1} zeta( 2*n ) / n!.

Original entry on oeis.org

2, 2, 2, 4, 1, 34, 4, 2, 3, 5, 1, 33, 2, 3, 1, 1, 12, 1, 20, 1, 9, 1, 2, 1, 4, 3, 1, 13, 1, 1, 2, 3, 1, 1, 8, 6, 1, 1, 3, 42, 1, 94, 1, 4, 2, 1, 1, 1, 7, 1, 1, 1, 16, 1, 25, 2, 1, 1, 29, 2, 3, 2, 3, 10, 6, 2, 1, 1, 1, 2, 1, 4, 2, 132, 1, 3, 1, 2, 8, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 45, 1, 2, 1, 5

Offset: 0

Frank Ellermann, Jul 13 2003

			2.40744655479032851470948665622302272558226649037984418869339833...

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

Decimal expansion is in A076813.

Offset changed by Andrew Howroyd, Aug 07 2024

cp's OEIS Frontend

A093720 Decimal expansion of Sum_{n >= 2} zeta(n)/n!.

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A080729 Decimal expansion of the infinite product of zeta functions for even arguments.

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A093721 Decimal expansion of Sum_{n>=1} zeta(2n)/(2n)!.

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A084862 Continued fraction expansion of Sum_{n>=1} zeta( 2*n ) / n!.

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