[seqfan] Re: An interesting sequence

[Peter Munn]

Arthur O'Dwyer:
> Is the substitution of "mean" for "sum" really a non-trivial operation,

I would have thought not. However, it is worth looking at the pink box
discussion in A360530's history, which is related. I'm not convinced that
the explanation that follows the formula in A360530 should be as short as
"(Lagrange)", but I will defer to better informed voices.

Could readers also note the related draft A362068, and avoid submitting a
duplicate.

Best Regards,

Peter

On Sun, April 16, 2023 6:02 pm, Arthur O'Dwyer wrote:
> (Adding participants-thus-far on the To: line.)
>
> It seems it has been proven ([1]) that all natural numbers above 245 are
> expressible as the sum of 5 distinct squares.
> This is pretty much what you need, since (n is the mean of 5 distinct
> squares) = (nx is the sum of 5 distinct squares).
> I'm not sure that [1] applies to nonzero squares specifically, but the
> existence of OEIS A004431–A004434 and the
> absence of a corresponding OEIS A004435 seems pretty conclusive
> circumstantial evidence.
>
> The integers' expressibility-as-sums-of-distinct-squares seems to have
> been
> pretty completely explored in OEIS already:
>
> https://oeis.org/A004431 -- numbers which are expressible as the sum of 2
> distinct nonzero squares
> https://oeis.org/A004432 -- numbers which are expressible as the sum of 3
> distinct nonzero squares
> https://oeis.org/A004433 -- numbers which are expressible as the sum of 4
> distinct nonzero squares
> https://oeis.org/A004434 -- numbers which are expressible as the sum of 5
> distinct nonzero squares
>
> Is the substitution of "mean" for "sum" really a non-trivial operation, or
> will this sequence be redundant with (readily derivable from) the above?
>
> https://math.stackexchange.com/a/4021787/
> https://math.stackexchange.com/a/1284951/
> https://math.stackexchange.com/a/852569/
> [1] — "Darstellung natürlicher Zahlen als Summe von Quadraten" (Franz
> Halter-Koch, 1982) http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4212.pdf
>
> -- Arthur
>
> On Sun, Apr 16, 2023 at 11:31AM Arthur O'Dwyer
> <arthur.j.odwyer at gmail.com>
> wrote:
>
>> On Sun, Apr 16, 2023 at 1:07AM Yifan Xie <xieyifan4013 at 163.com>
>> wrote:
>>
>>> a(n) is the smallest positive integer x such that n can be expressed as
>>> the arithmetic mean of x distinct nonzero squares, or 0 if x does not
>>> exist. Based on my calculation of a(1) to a(76) by hand, only a(2),
>>> a(3),
>>> a(6), a(8), a(12) are 0 and no terms are larger than 5.
>>> Please consider this sequence, and if possible, provide a program for
>>> me.
>>>
>>
>> I've given it a shot in C++.  It took me a few tries to notice all the
>> subtleties:
>> - distinct squares (so e.g. 3 = (1+4+4)/3, but we don't count that as a
>> solution)
>> - nonzero squares (so e.g. 2 = (0+4)/2, but we don't count that as a
>> solution)
>> But after that, it can do up to about n=50'000 in a few seconds.
>> Here are all the values
[...]