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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.
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%I #25 Feb 06 2022 02:48:56

%S 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,

%T 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,

%U 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

%N 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.

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

%H Sofia Kalpazidou, <a href="http://dx.doi.org/10.1016/0022-314X(88)90099-6">Khintchine's constant for Lüroth representation</a>, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.

%F U = exp(A085361).

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

%F Equals e * A085291. - _Amiram Eldar_, Jun 27 2021

%F Equals 1/A242624. - _Amiram Eldar_, Feb 06 2022

%e 2.200161058099026553194557866559944872685662324752723888723145117763169...

%p evalf(exp(Sum((Zeta(n+1)-1)/n, n=1..infinity)), 120); # _Vaclav Kotesovec_, Dec 11 2015

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

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

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jul 15 2014

%E Corrected by _Vaclav Kotesovec_, Dec 11 2015