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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
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
Sequence in context: A308756 A225371 A088310 * A005154 A074056 A206158
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.)