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

[M. F. Hasler]

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