[seqfan] Re: An interesting sequence

[israel at math.ubc.ca]

It's worth noting that the average of x distinct nonzero squares 
>= (x+1)*(2*x+1)/6, so x <= (sqrt(1+48*n)-1)/4.  
Here's my Maple program:

T:= proc(x,m) 
# sums of x distinct squares in {1^2, ..., m^2}
option remember;
 if x = 0 then return {0} 
 elif m < x then return {}
 fi;
 procname(x,m-1) union map(`+`,procname(x-1,m-1),m^2)
end proc:

f:= proc(n) local S,X,t,x,xmax, tmax;
 xmax:= floor((sqrt(1+48*n)-1)/4);
 for x from 1 to xmax do
  tmax:= floor(sqrt(x*n));
  if member(n*x, T(x,tmax)) then return x fi
 od;
 0
end proc:

Here are the first 200 terms:

1, 0, 0, 1, 2, 0, 3, 0, 1, 2, 5, 0, 2, 3, 3, 1, 2, 3, 5, 2, 4, 3, 3, 5, 1, 
2, 3, 3, 2, 3, 3, 5, 5, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 5, 2, 3, 3, 5, 1, 2, 
3, 2, 2, 3, 3, 3, 3, 2, 4, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 2, 2, 3, 
5, 4, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3, 3, 2, 2, 3, 3, 4, 3, 3, 5, 2, 3, 3, 1, 
2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 2, 3, 3, 3, 1, 2, 3, 3, 2, 
3, 3, 5, 3, 2, 3, 5, 4, 3, 3, 2, 2, 3, 3, 3, 4, 3, 3, 1, 2, 2, 3, 2, 2, 3, 
3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 4, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 
5, 3, 2, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 3, 3, 5, 2, 2, 3, 1, 2, 3, 3, 2

Cheers,
Robert

On Apr 16 2023, Fred Lunnon wrote:

>Ouch --- thanks.
>Now that blunder is fixed, my brute-force Magma lash-up finds
>
>    x = 5  for  n  in  {5, 19, 24, 32, 33, 44, 48, 76} ;
>
>    76 = ( 3^2 + 5^2 + 9^2 + 11^2 + 12^2 )/5 .
>
>No further zeros up to next standout case at  n = 96 .
>
>WFL
>
>
>On Sun, Apr 16, 2023 at 11:23AM <jens at voss-ahrensburg.de> wrote:
>
>>
>> a(11) = 5 since 11 = (25 + 16 + 9 + 4 + 1) / 5.
>>
>> Am 2023-04-16 12:07, schrieb Fred Lunnon:
>> > << only a(2), a(3), a(6), a(8), a(12) are 0 >>
>> >
>> > What about  n = 11  ?!     WFL
>> >
>> >
>> >
>> > On Sun, Apr 16, 2023 at 6:07AM Yifan Xie <xieyifan4013 at 163.com> wrote:
>> >
>> >> Hi,
>> >> 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.
>> >>
>> >> Best regards,
>> >> Yifan Xie (xieyifan4013 at 163.com)
>>
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
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