|
| |
|
|
A134473
|
|
a(n) = the smallest positive integer such that sum{k=1 to n} 1/a(k) is <= product{j=1 to 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 to n} 1/a(k) increases, but is bounded from above (by the product). While product{j=1 to 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 .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 to n-1} 1/(1 +1/a(j)) and y = sum{k=1 to n-1} 1/a(k), then a(n) = ceiling[(1 + y + sqrt((y-1)^2 + 4x))/(2(x-y))].
|
|
|
EXAMPLE
|
sum{k=1 to 2} 1/a(k) = 3/5 and product{j=1 to 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) ; [From R. J. Mathar, Jul 20 2009]
|
|
|
CROSSREFS
|
Cf. A134474, A134475, A134476, A134477.
Sequence in context: A193482 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
|
| |
|
|
