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

David Corneth davidacorneth at gmail.com
Mon Aug 3 10:33:28 CEST 2020


P.s. I see now here:
https://oeis.org/wiki/Index:_Instructions_For_Updating_Index_to_OEIS   that
adding a new index is for experts only so If someone else could do that I'm
happy to help editing from there.

On Mon, Aug 3, 2020 at 10:31 AM David Corneth <davidacorneth at gmail.com>
wrote:

> 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
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>


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