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A134473
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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.
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5
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2, 10, 265, 186534, 39698716206, 9708281043219621795399, 485147416562376967927656482516055847985046599, 261312356099926248292437979417147998592741394591619008401746229884484893481820640113595606
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OFFSET
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1,1
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COMMENTS
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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?
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LINKS
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FORMULA
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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))).
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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))]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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