[seqfan] Re: 448 ways

[jean-paul allouche]

Could the expression "nonnegative squares" be an awkward way
of saying "squares of nonnegative numbers"?
(cf. the number of decompositions of an integer into a sum
of four squares n = x^2 + y^2 + z^2+t^2, where you count
(x)^2 and (-x)^2 as giving distinct or not distinct decompositions).
Of course being zero or not zero is a different story.

jpa



Le 19/11/17 à 17:33, Allan Wechsler a écrit :
> At least in https://oeis.org/A295159, the sequence we started with, the
> word "nonnegative" could be removed, because all integer squares are
> nonnegative. But it should not be changed to "nonzero". For example
> A295159(1) = 0; that is, the smallest number that is the sum of five
> nonnegative squares in exactly one way is 0, because 0 = 0+0+0+0+0. But 0
> is not the sum of five nonzero squares in any way whatsoever. Similarly,
> A295159(3) = 13, because 13 = 9 + 4 + 0 + 0 + 0 = 9 + 1 + 1 + 1 + 1 = 4 + 4
> + 4 + 1 + 0.  If the word "nonnegative" in the title were changed to
> "nonzero", two of these representations would be invalid.
>
> A295159(n) is the smallest k such that A000174(k) = n. If we replace
> "nonnegative" with "nonzero", we would need to replace A000174 with
> https://oeis.org/A025429.
>
>
>
> On Sun, Nov 19, 2017 at 10:36 AM, Neil Sloane <njasloane at gmail.com> wrote:
>
>> Could the explanation (why on earth would someone say "nonnegative
>> square"?)
>> be that they meant to say "nonzero sequence"?  It's not a typo, it's a
>> braino.    So the fix would be to replace
>> "nonnegative square" with "nonzero square" ??
>>
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
>>
>> On Sat, Nov 18, 2017 at 12:10 PM, Peter Munn <techsubs at pearceneptune.co.uk
>> wrote:
>>
>>> Dear seqfans,
>>>
>>> On Thu, November 16, 2017 6:09 pm, Allan Wechsler wrote:
>>>> Very interesting! So we need a new sequence, "numbers N for which no K
>>> is
>>>> the sum of five squares in exactly N ways", of which 448 will be the
>>> leading entry.
>>>
>>> Another sequence in which 448 would be the first term is "least m for
>>> which no K is the sum of n squares in exactly m ways", which would have
>> an
>>> offset for n of 5.  Maybe it would be easier to calculate sufficient
>> terms
>>> for this?
>>>
>>> By the way, how do others see the "nonnegative" qualification for
>> "square"
>>> in the name of A295159?
>>>
>>> 19 sequences showed up in the OEIS search as having "nonnegative square"
>>> in their name (when excluding those also having "root" in the name), but
>>> only 7 of them date from before this month.
>>>
>>> Four of these, A229062, A073613, A160053 and A180161, do so with clear
>>> reference to A001481, which no longer has the qualification in its own
>>> name, "Numbers that are the sum of 2 squares."  Another, A230314,
>> "Numbers
>>> that are simultaneously the sum of two nonnegative squares and the sum of
>>> two nonnegative cubes", benefits by pointing out a symmetry.
>>>
>>> Of the remaining two, A006431, "Numbers that have a unique partition into
>>> a sum of four nonnegative squares" is clearly the one related to A295159.
>>> The other, A166498, has had its "nonnegative" deleted following my
>>> questioning it when making a correction.
>>>
>>> Best Regards,
>>>   Peter Munn
>>>> On Thu, Nov 16, 2017 at 7:14 AM, Giovanni Resta
>>>> <giovanni.resta at iit.cnr.it>
>>>> wrote:
>>>>> On 11/16/2017 03:51 AM, israel at math.ubc.ca wrote:
>>>>> To make that lower bound 448 would require x >= 2339578, and that's
>> too
>>>>>> big for me to search. But maybe better lower bounds are known.
>>>>> I checked all the x between 25000 and 2339600 and all have at least
>> 449
>>> decompositions as a sum of five squares.
>>>>> Giovanni
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>>>
>>>
>>>
>>>
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