A134473 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.

2, 10, 265, 186534, 39698716206, 9708281043219621795399, 485147416562376967927656482516055847985046599, 261312356099926248292437979417147998592741394591619008401746229884484893481820640113595606

Offset

1,1

Comments

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?

Links

Table of n, a(n) for n=1..8.

Formula

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))).

Example

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.

Maple

Digits := 220 ; A134473 := proc(n) option remember ; local su, mu ;
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:
seq(A134473(n), n=1..9) ; # R. J. Mathar, Jul 20 2009

Mathematica

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))]]];
Table[a[n], {n, 1, 8}] (* Jean-François Alcover, Sep 26 2022 *)

Crossrefs

Cf. A134474, A134475, A134476, A134477.

Sequence in context: A308756 A225371 A088310 * A005154 A074056 A206158

Adjacent sequences: A134470 A134471 A134472 * A134474 A134475 A134476

Keyword

nonn

Author

Leroy Quet, Oct 27 2007

Extensions

More terms from R. J. Mathar, Jul 20 2009

Status

approved