%I #19 Sep 26 2022 06:16:19
%S 2,10,265,186534,39698716206,9708281043219621795399,
%T 485147416562376967927656482516055847985046599,
%U 261312356099926248292437979417147998592741394591619008401746229884484893481820640113595606
%N a(n) is the smallest positive integer such that Sum_{k=1..n} 1/a(k) <= Product_{j=1..n} 1/(1 + 1/a(j)), for every positive integer n.
%C Sum_{k=1..n} 1/a(k) increases, but is bounded from above (by the product), while Product_{j=1..n} 1/(1 + 1/a(j)) decreases and is bounded from below (by the sum). The sum and the product then approach the same constant, which is approximately 0.6037789..., if their difference approaches 0. Does this constant have a closed form in terms of known constants, if the constant exists?
%F For n >= 2, if x = Product_{j=1..n-1} 1/(1 + 1/a(j)) and y = Sum_{k=1..n-1} 1/a(k), then a(n) = ceiling((1 + y + sqrt((y-1)^2 + 4x))/(2(x-y))).
%e Sum_{k=1..2} 1/a(k) = 3/5 and Product_{j=1..2} 1/(1 + 1/a(j)) = 20/33. For m = any positive integer <= 264, 3/5 + 1/m is > 20/(33*(1 + 1/m)). But if m = 265, then 3/5 + 1/m = 32/53 is <= 20/(33*(1 + 1/m)) = 2650/4389. So a(3) = 265.
%p Digits := 220 ; A134473 := proc(n) option remember ; local su,mu ;
%p if n = 1 then 2; else su := add(1/procname(k),k=1..n-1) ; mu := mul(1/(1+1/procname(j)), j=1..n-1) ; ceil( (1+su+sqrt((su-1)^2+4*mu))/2/(mu-su) ) ; fi; end:
%p seq(A134473(n), n=1..9) ; # _R. J. Mathar_, Jul 20 2009
%t a[n_] := a[n] = If[n == 1, 2, With[{x = Product[1/(1+1/a[j]), {j, 1, n-1}], y = Sum[1/a[j], {j, 1, n-1}]}, Ceiling[(1+y+Sqrt[(y-1)^2+4x])/(2(x-y))]]];
%t Table[a[n], {n, 1, 8}] (* _Jean-François Alcover_, Sep 26 2022 *)
%Y Cf. A134474, A134475, A134476, A134477.
%K nonn
%O 1,1
%A _Leroy Quet_, Oct 27 2007
%E More terms from _R. J. Mathar_, Jul 20 2009