1,1

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?

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))].

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.

Cf. A134474, A134475, A134476, A134477.

Sequence in context: A037267 A001528 A088310 this_sequence A005154 A074056 A003047

Adjacent sequences: A134470 A134471 A134472 this_sequence A134474 A134475 A134476

more,nonn

Leroy Quet (qq-quet(AT)mindspring.com), Oct 27 2007