|
| |
|
|
A231692
|
|
Define a sequence of rationals by f(0)=0, thereafter f(n)=f(n-1)-1/n if that is >= 0, otherwise f(n)=f(n-1)+1/n; a(n) = numerator of f(n).
|
|
3
|
|
|
|
0, 1, 1, 1, 5, 13, 1, 27, 19, 451, 199, 4709, 2399, 3467, 29207, 5183, 55411, 221267, 300649, 1628251, 5508127, 259001, 762881, 6460903, 5694791, 11476403, 27820203, 326206681, 5151783667, 69088293143, 146724611903, 2219373406193, 8951357840311, 4575492601111, 328329280711, 4454145077671
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
It is conjectured that the terms of the {f(n)} sequence are distinct.
If that is true, then the {f(n)} sequence is a fractional analog of Recamán's sequence A005132.
The denominators of {f(n)} form A231693 (a non-monotonic sequence).
Here is a proof that the f(n) are distinct: Suppose not. Then the difference between the terms (which is zero) is a number of the form +- 1/(a+1) +- 1/(a+2) +- 1/(a+3) +- ... +- 1/(a+n).
Consider any harmonic sum
S = +- 1/(a+1) +- 1/(a+2) +- 1/(a+3) +- ... +- 1/(a+n)
where one puts any sign on any term, and there is at least one term. Let G be the LCM of the denominator(s). Then for any denominator D, G/D is an integer, and G*S = sum(+- G/D_i) is an integer. Let E be the highest power of two that divides G. Then there is only one multiple of E among the denominators. (If there were two, they would be consecutive multiples of E, and one would be divisible by 2*E.) Call that denominator F. So (+- G/F) is an odd integer, and for all other denominators D, (+- G/D) is an even integer. Therefore G*S is odd, therefore not zero, so S is not zero. (End)
|
|
|
REFERENCES
|
David Wilson, Posting to Sequence Fans Mailing List, Nov 14 2013.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
0, 1, 1/2, 1/6, 5/12, 13/60, 1/20, 27/140, 19/280, 451/2520, 199/2520, 4709/27720, ...
|
|
|
MAPLE
|
f:=proc(n) option remember;
if n=0 then 0 elif
f(n-1) >= 1/n then f(n-1)-1/n else f(n-1)+1/n; fi; end;
|
|
|
PROG
|
(PARI) s=0; vector(30, n, numerator(s-=(-1)^(n*s<1)/n)) \\ - M. F. Hasler, Nov 15 2013
(Haskell)
a231692_list = map numerator $ 0 : wilson 1 0 where
wilson x y = y' : wilson (x + 1) y'
where y' = y + (if y < 1 % x then 1 else -1) % x
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,frac
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
