A355499 Decimal expansion of Product_{k>=1} (k - 2/3)^(1/(k - 2/3)) / k^(1/k).

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

Offset: 0

Author

Vaclav Kotesovec, Jul 04 2022

Keywords

• nonn
• cons

Examples

			0.0413062412559336395283825210006728108317741296744868855779544405463319...
		

Links

• R. W. Gosper, Some identities, (d162).

Crossrefs

Cf. A001620, A115522, A355500.

Programs

Maple

evalf((3^(1/4) * exp(-gamma/2) * GAMMA(1/3)^3 / (4*Pi^2))^(Pi/sqrt(3)) / 3^(3*(log(3) + 2*gamma)/4), 120);

Mathematica

Join[{0},RealDigits[(3^(1/4) * Exp[-EulerGamma/2] * Gamma[1/3]^3/4/Pi^2)^ (Pi/Sqrt[3])/3^(3*(Log[3] + 2*EulerGamma)/4), 10, 120][[1]]]

PARI

default(realprecision, 200); exp(sumpos(n=1, log(n - 2/3)/(n - 2/3) - log(n)/n))

Formula

Equals (3^(1/4) * exp(-gamma/2) * Gamma(1/3)^3 / (4*Pi^2))^(Pi/sqrt(3)) / 3^(3*(log(3) + 2*gamma)/4), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.