

A134473


a(n) = 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.


5



2, 10, 265, 186534, 39698716206, 9708281043219621795399, 485147416562376967927656482516055847985046599, 261312356099926248292437979417147998592741394591619008401746229884484893481820640113595606
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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..n1} 1/(1 + 1/a(j)) and y = Sum_{k=1..n1} 1/a(k), then a(n) = ceiling((1 + y + sqrt((y1)^2 + 4x))/(2(xy))).


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..n1) ; mu := mul(1/(1+1/procname(j)), j=1..n1) ; ceil( (1+su+sqrt((su1)^2+4*mu))/2/(musu) ) ; fi; end: seq(A134473(n), n=1..9) ; # R. J. Mathar, Jul 20 2009


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



