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).

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

Offset: 0

Author

N. J. A. Sloane, Nov 15 2013, Nov 16 2013

Keywords

• nonn
• frac

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).

From Don Reble, Nov 16 2013: (Start)

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)

Examples

			0, 1, 1/2, 1/6, 5/12, 13/60, 1/20, 27/140, 19/280, 451/2520, 199/2520, 4709/27720, ...
		

References

• David Wilson, Posting to Sequence Fans Mailing List, Nov 14 2013.

Links

• David W. Wilson and Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A231693, A005132.

Programs

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
-- Reinhard Zumkeller, Nov 16 2013

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;

Mathematica

Numerator[FoldList[# + (-1)^Boole[#*#2 >= 1]/#2 &, Range[0, 35]]] (* Paolo Xausa, Aug 16 2025 *)

PARI

s=0;vector(30,n,numerator(s-=(-1)^(n*s<1)/n)) \\ - M. F. Hasler, Nov 15 2013