[seqfan] Re: Numbers that are the sum of x nonzero y-th powers

[David Corneth]

I have for each of


A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325
(3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3),
A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335
(12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4),
A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344
(10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5),
A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8,
5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5),
A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6,
6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6),
A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371
(4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7),
A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380
(2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8),
A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390
(12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9),
A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399
(10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3,
10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10),
A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10),
A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11),
A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820
(9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5,
2).

a 10000-term b-file ready. That's quite doable. Then I'll cook up 3
examples and a comment saying something like: as the order of addition
doesn't matter we can assume the terms are in nondecreasing order.
Do we add the x-refs to each of these? Or maybe a wikipage? I don't know
how to set up a wikipage so If someone could help there that'd be great.




On Sat, Aug 1, 2020 at 7:26 PM Marc LeBrun <mlb at well.com> wrote:

> To actually fix it, someone could either
>    1. change the title to "numbers that are a sum of 7 numbers in 1
> through 5 raised to the 8th power"
> or 2. change the sequence values so they actually include all the numbers
>
> (I recommend #2, although it's more work).
>
> At the very least, some comment should be quickly added to warn users that
> the sequence as it stands is incorrect.
>
>
> > On Jul 31, 2020, at 4:11 PM, David Corneth <davidacorneth at gmail.com>
> wrote:
> >
> > Hi All,
> >
> > I just came across A003385 which is called:
> > Numbers that are the sum of 7 nonzero 8th powers.
> >
> > And there is a whole series of such sequences, starting at about A003325
> > going through A003400.
> >
> > I think at least some but still quite many need the following fixes.
> > The most important ones are b-files and progs.
> > A003385 has a b-file of 3432 terms, the largest of them being: 117440512.
> > However, up to that bound, 3319 terms are missing.
> > Possibly it's due to the prog at hand, or at least the idea.
> > The prog there is in Mathematica:
> > Total/@Tuples[Range[5]^8, 7]//Union
> > And such progs are all arround those sequences.
> > Now I don't know Mathematica quite well but I looked in the documentation
> > and range[5] gives the array [1, 2, 3, 4, 5] and each of these numbers is
> > going to be raised to the 8th power, giving:
> > [1, 128, 2187, 16384, 78125]. Then some tuples (5^7 of them) are made and
> > that makes a list of distinct terms. i.e. duplicates are removed, and
> there
> > will be a lot of duplicates; all permutations of a sorted array of 7
> > numbers each in 1 through 5 raised to the 8th power.
> >
> > The first missing term in the b-file is 43046727 which is 1^8 + 1^8  +
> 1^8
> > + 1^8  + 1^8  + 1^8  + 9^8.
> > Now a quick look shows me a lot of these sequences have such progs.
> > Another one is A003384 which has the Mathematica program:
> > Union[Total[#^8]&/@Tuples[Range[10], 6]]
> > same issue.
> > Some look better, like With[{upto=10000},
> > Select[Union[Total/@Tuples[Range[Floor[Surd[upto-4, 6]]]^6, 5]],
> #<=upto&]]
> > as in correct but perhaps inefficient.
> > in A003361:  Numbers that are the sum of 5 nonzero 6th powers.
> > At any rate, I think they need to be revised.
> >
> > Some lack a b-file and could have that.
> > In some sequences, data could be extended to the 260 chars.
> > And probably they could xref each other more.
> > And an example or two would be nice.
> >
> > What's a good approach here? What changes should actually be made? Who
> can
> > help?
> >
> > Best,
> > David
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
>
>
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