A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

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

Offset: 1

Author

Jean-François Alcover, Jul 15 2014

Keywords

• nonn
• cons

Comments

The geometric mean of the Yule-Simon distribution with parameter value 1 (A383855) approaches this constant. In general, the geometric mean of the Yule-Simon distribution approaches Product_{k>=2} k^(1/(p*Beta(k,p+1))). - Jwalin Bhatt, May 12 2025

Examples

			2.200161058099026553194557866559944872685662324752723888723145117763169...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

Links

• Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.

Crossrefs

Cf. A002210, A085291, A085361, A242624, A244109(V), A131688(W), A383855.

Programs

Maple

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

Mathematica

Exp[NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}]] (* Vaclav Kotesovec, Dec 11 2015 *)

Formula

Equals exp(A085361).

U*V*W = 1, where V is A244109 and W is A131688.

Equals e * A085291. - Amiram Eldar, Jun 27 2021

Equals 1/A242624. - Amiram Eldar, Feb 06 2022

Extensions

Corrected by Vaclav Kotesovec, Dec 11 2015