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

[David Corneth]

Thanks Neil and Maximilian. I like to read that the effort helps :)
I'll try to set up a wikipage but this would be my first so please
supervise and advise if possible.

To make xrefs, in any format desired probably, I'll list the sequences in a
(PARI) vector format in order of appearance in OEIS (including 3386 now and
3325 fixed) here:
[[404, 2, 2], [3325, 2, 3], [3336, 2, 4], [3347, 2, 5], [3358, 2, 6],
[3369, 2, 7], [3380, 2, 8], [3391, 2, 9], [4802, 2, 10], [4813, 2, 11],
[408, 3, 2], [3072, 3, 3], [3337, 3, 4], [3348, 3, 5], [3359, 3, 6], [3370,
3, 7], [3381, 3, 8], [3392, 3, 9], [4803, 3, 10], [4814, 3, 11], [414, 4,
2], [3327, 4, 3], [3338, 4, 4], [3349, 4, 5], [3360, 4, 6], [3371, 4, 7],
[3382, 4, 8], [3393, 4, 9], [4804, 4, 10], [4815, 4, 11], [47700, 5, 2],
[3328, 5, 3], [3339, 5, 4], [3350, 5, 5], [3361, 5, 6], [3372, 5, 7],
[3383, 5, 8], [3394, 5, 9], [4805, 5, 10], [4816, 5, 11], [3329, 6, 3],
[3340, 6, 4], [3351, 6, 5], [3362, 6, 6], [3373, 6, 7], [3384, 6, 8],
[3395, 6, 9], [4806, 6, 10], [4817, 6, 11], [3330, 7, 3], [3341, 7, 4],
[3352, 7, 5], [3363, 7, 6], [3374, 7, 7], [3385, 7, 8], [3396, 7, 9],
[4807, 7, 10], [4818, 7, 11], [3331, 8, 3], [3342, 8, 4], [3353, 8, 5],
[3364, 8, 6], [3375, 8, 7], [3386, 8, 8], [3397, 8, 9], [4808, 8, 10],
[4819, 8, 11], [3332, 9, 3], [3343, 9, 4], [3354, 9, 5], [3365, 9, 6],
[3376, 9, 7], [3387, 9, 8], [3398, 9, 9], [4809, 9, 10], [4820, 9, 11],
[3333, 10, 3], [3344, 10, 4], [3355, 10, 5], [3366, 10, 6], [3377, 10, 7],
[3388, 10, 8], [3399, 10, 9], [4810, 10, 10], [4821, 10, 11], [3334, 11,
3], [3345, 11, 4], [3356, 11, 5], [3367, 11, 6], [3378, 11, 7], [3389, 11,
8], [4800, 11, 9], [4811, 11, 10], [4822, 11, 11], [3335, 12, 3], [3346,
12, 4], [3357, 12, 5], [3368, 12, 6], [3379, 12, 7], [3390, 12, 8], [4801,
12, 9], [4812, 12, 10], [4823, 12, 11]]

and ordered by n-th power here:
If that order is preferred.
I used the former order to make the xrefs using this snippet (specifically
for this family, though Anumber could be used in general in PARI)
xrefline(v) = {
res = "Cf. ";
for(i = 1, #v,
res = concat([res, Anumber(v[i][1]), " (", v[i][2], ", ", v[i][3], "), "]);
); concat(res, ".")
}

Anumber(n) = {res = "A"; for(i = #digits(n) + 1, 6, res = concat(res, 0);
); return(concat(res, n))}

This takes a vector of the form above and prints them in that format I gave
in the order of the vector.

I'll proceed with the 'lighter' editing (that's change name, extend data
(if needed), put examples and kw easy for now. I'm happy to come back to
earlier sequences I touched here if a global search isn't enough.

Best,
David

On Mon, Aug 3, 2020 at 8:45 AM M. F. Hasler <oeis at hasler.fr> wrote:

> On Mon, Aug 3, 2020 at 6:20 AM David Corneth <davidacorneth at gmail.com>
> wrote:
>
> > It depends on what we say is ready. I like the suggestion of designating
> > one sequence as a 'guide sequence'. And have any others I look at say
> that
> > that sequence has that information, like which are related sequences.
> Would
> > that be okay?
> >
>
> I also like this. If we give as XREFS only the lower and upper end of
> contiguous ranges this does the job, i.e., e.g.,
> Cf. numbers that are the sum of x nonzero y-th powers:
>   A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
>   A003325 (2, 3), A003072 (3, 3), A003327 .. A004823 (x = 2 .. 12, y = 3 ..
> 11)."
>
> but in the case at hand one could put intermediate "guides", viz:
> ...
>   A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
>   A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
>   A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
>   A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
>   A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).
>
> Now I think I removed the incorrect programs. And I think there are no more
> > incorrect b-files in those sequences.
> >
>
> Great job, thanks David!
>
>
> > Not all sequences have the same format for the name. For example we have
> > "Sum of 11 positive 11th powers." (A004822)
> > and Numbers n which are the sum of 3 nonzero 4th powers. (A003337)
> >
>
> No need to introduce an unused variable "n" here, personally I'd prefer:
>   Numbers which are the sum of 3 nonzero 4th powers.
>
> The first variant should have at least "Sums ..." to be "grammatically"
> and/or semantically correct, I think.
>
> And I have 3 examples prepared for all those sequences, automated.
> >
>
> That's also great. (Although a link to a relevant "guide" sequence with
> carefully chosen and ~worded examples could be sufficient here.)
>
> I'm adding PARI code to those sequence that don't already have.
> In the process, I double check DATA and complete it to the recommended
> length.
> Concerning "terms in nondecreasing order", it's a comment which is only
> relevant for the programs, not for the sequence itself, where it can be a
> bit confusing at first sight.
> Also, it can be faster to produce the terms through built-in mechanisms
> (such as: exponents in (Sum_{n>=1} X^n^y)^x) without considering the order
> of the terms, rather than to do it by hand in order to avoid the
> "duplicates".
> If the first gives 10^4 terms within milliseconds and with short & simple
> code, that should not be disfavored wrt lengthier,  more sophisticated but
> possibly error-prone programs.
>
> Maximilian
>
> --
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>