terms lack

[Dean Hickerson]

Mostly to Yasutoshi Kohmoto <zbi74583 at boat.zero.ad.jp>:

> Recently I updated more terms of A054572, which are calculated by
> Richard Mathar.
> And Neil claimed or got angry.
> He said, "You are red card. Don't submit a wrong sequences"
> I am exaggerating a little.
> These words are not exactly what he wrote.
>
> To Neil
> They are correct, not "wrong".
> It is only not complete.

The terms that are listed in a sequence should be the start of the sequence.
If there are missing terms in between the listed ones, then the sequence
entry is not just incomplete, it's wrong.

If you've computed some terms of a sequence, but you're not sure if there
are smaller ones that you've missed, then you should only include the ones
that you're sure are at the start of the sequence.  You can add a comment
about the larger terms, stating that it's not known if there are smaller ones.
For example, the "EXTENSIONS" lines for the sequence A000043 (Mersenne
exponents) mention some recently discovered terms.  But there might be
some smaller ones, so these are not included in the sequence.

> I defined many "divisor" functions and calculated many Amicable Numbers and
> Perfect Numbers using those functions.
> And I submitted them as sequences to  OEIS.
>
> http://www.research.att.com/~njas/sequences/?q=kohmoto+amicable&sort=0&fmt=0&language=english&go=Search
>
> http://www.research.att.com/~njas/sequences/?q=kohmoto+perfect&sort=0&fmt=0&language=english&go=Search
>
> I suppose that they are almost complete but some of them might not be
> complete.

I've found missing terms for several of your sequences:


Sequences A045613 and A045614 ("Super Unitary Amicable Number") consist
of pairs of distinct numbers a and b such that

    usigma(usigma(a)) = usigma(usigma(b)) = a+b.

(The definitions don't make this clear, but A045613 contains the smaller
members of the pairs and A045614 contains the larger members.  Also they're
sorted by their smallest members, so 155 comes before 142 in A045614.)  The
first 3 pairs that you've listed are (105, 155), (110, 142) and (2145, 3055).
But you've missed several smaller ones:

    (33, 35), (208, 224), (268, 272), (455, 601), (695, 1033), (812, 956),
    (1609, 1847), (1808, 2512), (1892, 3004), (1913, 2407), (2096, 2224)


A051594 and A051595 are "(-1)sigma amicable numbers".  (Again, the definition
doesn't make it clear that A051595 has the smaller terms and A051594 has the
larger terms.)  The first pair that you've listed is (1969706592, 2236072608).
But there are many smaller ones, starting with:

    (429552, 466200), (497808, 604656), (800496, 1103760).


A036471-A036474 are defined as "Amicable quadruple: 4 different numbers
which satisfy sigma(a)=sigma(b)=sigma(c)=sigma(d)=a+b+c+d".  The first
quadruple that you've listed has smallest element
342151462276356306033089201934180, but here's a much smaller one:

    (3270960, 3361680, 3461040, 3834000)

(I think that's the smallest one, but I could be wrong.)  If someone can
determine the first several terms of this sequence, then it should be
edited; otherwise it should be deleted.


In some of your sequences, I wasn't able to find any smaller terms, but the
ones that you've listed are so large that I'm not at all confident that they
are the smallest ones.  For example, A038362 and A038363 are defined as
"A Rational Amicable Number: two different numbers a, b which satisfies the
following equation: sigma(a)=sigma(b)=(a+b)^3/(a^2+b^2)", and the smallest
pair that you've included is:

    (26403469440047700, 30193441130006700)

Do you have any reason to think that that's the smallest such pair?  If not,
and if you can't determine what the first few terms really are, then the
sequence should be deleted.


I've only looked at a few of your sequences, but that's enough to convince
me that they should all be checked.

Neil, I'll submit corrections for A045613, A045614, A051594, and A051595,
but I'm not going to work on the others.  Maybe they should be deleted or
marked as "probation" until the author or someone else fixes them.

Dean Hickerson
dean at math.ucdavis.edu