A131709

[N. J. A. Sloane]

Regarding sequence A134473:

>%I A134473
>%S A134473 2,10,265,186534
>%N A134473 a(n) = the smallest positive integer
such that sum{k=1 to n} 
>1/a(k) is <= product{j=1 to n} 1/(1 +1/a(j)),
for every positive integer n.
>%C A134473 sum{k=1 to n} 1/a(k)  increases, but
is bounded from above (by 
>the product). While product{j=1 to 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
..6037789..., if their difference 
>approaches 0. Does this constant have a closed
form in terms of known 
>constants, if the constant exists?
>%F A134473 For n >= 2,
>if x = product{j=1 to n-1} 1/(1 +1/a(j)), and y
= sum{k=1 to n-1} 1/a(k), 
>then
>a(n) = ceiling[(1 + y + sqrt((y-1)^2 +
4x))/(2(x-y))].
>%e A134473 sum{k=1 to 2} 1/a(k) = 3/5, and
product{j=1 to 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.
>%Y A134473 A134474,A134475,A134476,A134477
>%O A134473 1
>%K A134473 ,more,nonn,


I haven't thought too hard about this, but it
seems that it would be easy to prove that the sum
and the product approach the same constant. (The
two limits are mighty close, whatever the
situation.)

Call the sum limit S and the product limit P.

In any case, do the constants S and P
(.6037789... = the continued fraction
[0;1,1,1,1,9,1,65,....]) have (a) closed form(s)
in terms of known constants?


Thanks,
Leroy Quet


__________________________________________________
Do You Yahoo!?
Tired of spam?  Yahoo! Mail has the best spam protection around 
http://mail.yahoo.com