[seqfan] Re: Reciprocal Recaman

[franktaw at netscape.net]

A proof might be possible along the following lines:

Any match r(j) = r(k) must have P(n) <= j,k < P(n+1), for some n, where 
P(n) is the nth prime. (All terms P(n) <=j < 2*P(n) must have P(n) in 
the denominator, and P(n+1) < 2*P(n). Of course P(n+1) cannot be in the 
denominator for j < P(n+1); thus P(n) is the largest prime in the 
denominator.) The problem then becomes showing that no +|- 1/(j+1) +|- 
1/(j+2) +|- ... +|- 1/k can sum to zero when j and k are thus bounded 
close together. I don't have any idea how to do that, but perhaps 
someone else can help.

Franklin T. Adams-Watters

-----Original Message-----
From: David Wilson <davidwwilson at comcast.net>

Per the comment on A231693:

%C Since the terms are necessarily distinct (the denominators are
nondecreasing), this is a fractional analog of Recaman's sequence 
A005132.

I believe that all the terms of the reciprocal Recaman are distinct,
however, the reasoning above is insufficient, and incorrect.
If the denominators were nondecreasing, values with the same denominator
could still be equal.
But in fact, the denominators are not nondecreasing, e.g. a(18) < a(17).

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil
> Sloane
> Sent: Friday, November 15, 2013 9:07 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Reciprocal Recaman
>
> Oops! The denominators are A231693...
>
>
> On Fri, Nov 15, 2013 at 7:34 PM, <franktaw at netscape.net> wrote:
>
> > I would be surprised if they are.
> >
> > Franklin T. Adams-Watters
> >
> >
> > -----Original Message-----
> > From: David Wilson <davidwwilson at comcast.net>
> > To: 'Sequence Fanatics Discussion list' <seqfan at list.seqfan.eu>
> > Sent: Fri, Nov 15, 2013 6:04 pm
> > Subject: [seqfan] Re: Reciprocal Recaman
> >
> >
> > Do we know for sure that the denominators are A002805?
> >
> >  -----Original Message-----
> >> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of 
Neil
> >> Sloane
> >> Sent: Friday, November 15, 2013 5:57 AM
> >> To: Sequence Fanatics Discussion list
> >> Subject: [seqfan] Re: Reciprocal Recaman
> >>
> >> I added the numerators of f(n) as A231692. The denominators are
> >>
> > A002805.
> >
> >> I also added this to the Index to Fractions on the Wiki side.
> >> Perhaps someone else (Don?) could add Don's sequence.
> >> Neil
> >>
> >>
> >> On Fri, Nov 15, 2013 at 4:51 AM, Don Reble <djr at nk.ca> wrote:
> >>
> >> > f(0) = 0
> >> >> f(n) = f(n-1) - 1/n if >= 0, else f(n-1) + 1/n.
> >> >>
> >> >> For which n do we have f(n-2) > f(n-1) > f(n)?
> >> >>
> >> >
> >> >    I get
> >> >
> >> >    3 6 13 34 91 264 783 2342 7013 21030 63079 189236 567709 
1703124
> >> >    5109367 15328088 45984249
> >> >
> >> >    Since odds and evens alternate, I conclude that no term is 
triple
> >> >    the previous.
> >> >
> >> > --
> >> > Don Reble  djr at nk.ca
> >> >
> >> >
> >> >
> >> > _______________________________________________
> >> >
> >> > Seqfan Mailing list - http://list.seqfan.eu/
> >> >
> >>
> >>
> >>
> >> --
> >> Dear Friends, I have now retired from AT&T. New coordinates:
> >>
> >> Neil J. A. Sloane, President, OEIS Foundation
> >> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> >> Also Visiting Scientist, Math. Dept., Rutgers University, 
Piscataway,
> >>
> > NJ.
> >
> >> Phone: 732 828 6098; home page: http://NeilSloane.com
> >> Email: njasloane at gmail.com
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
> >
> > _______________________________________________
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> >
> > _______________________________________________
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>
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, 
NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/


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