cp's OEIS Frontend

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.

%I A245254 #36 May 16 2025 00:49:50
%S A245254 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 A245254 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 A245254 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 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.
%C A245254 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
%D A245254 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.
%H A245254 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 A245254 Equals exp(A085361).
%F A245254 U*V*W = 1, where V is A244109 and W is A131688.
%F A245254 Equals e * A085291. - _Amiram Eldar_, Jun 27 2021
%F A245254 Equals 1/A242624. - _Amiram Eldar_, Feb 06 2022
%e A245254 2.200161058099026553194557866559944872685662324752723888723145117763169...
%p A245254 evalf(exp(Sum((Zeta(n+1)-1)/n, n=1..infinity)), 120); # _Vaclav Kotesovec_, Dec 11 2015
%t A245254 Exp[NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}]] (* _Vaclav Kotesovec_, Dec 11 2015 *)
%Y A245254 Cf. A002210, A085291, A085361, A242624, A244109(V), A131688(W), A383855.
%K A245254 nonn,cons
%O A245254 1,1
%A A245254 _Jean-François Alcover_, Jul 15 2014
%E A245254 Corrected by _Vaclav Kotesovec_, Dec 11 2015