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

David Corneth davidacorneth at gmail.com
Sat Aug 1 01:11:55 CEST 2020

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


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