[seqfan] Re: Reciprocal Recaman

Don Reble djr at nk.ca
Sat Nov 16 07:05:23 CET 2013

> Did you use integer or double precision arithmetic to get these values?

    Exact rational arithmetic.
    For those who want to double-check, my 45984249th sum is
    with 19971600 digits elided from each.

> Do we know for sure that the denominators are A002805?

> They differ from a(20) on....

    Yes, but then they agree at a(28)-a(32), a(55)-a(64), a(88)-a(90),
    a(125)-a(142), ...

> I believe that all the terms of the reciprocal Recaman are
> distinct...

    Me, too. 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 g.c.d. 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 which 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, not zero, and S
    is not zero.

Don Reble  djr at nk.ca

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