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 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. 5
 2, 10, 265, 186534, 39698716206, 9708281043219621795399, 485147416562376967927656482516055847985046599, 261312356099926248292437979417147998592741394591619008401746229884484893481820640113595606 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified April 21 10:04 EDT 2024. Contains 371852 sequences. (Running on oeis4.)