unsquares in A004215

[John Conway]

On Mon, 2 Nov 1998, Wouter Meeussen wrote:

> hi,
> numbers that can not be written as x^2+y^2+z^2 with x,y,z integer (incl 0)
> 
> 7,15,23,28,31,39,47,55,60,63,71,79,87,92,95,103,111,112,119,124,127,135,143,
> 151,156,159,167,175,183,188,191,199,207,215,220,223,231,239,240,247,252,255,
> 263,271,279,284,287,295,303,311,316,319,327,335,343,348,351,359,367,368,375,
> 380,383,391,399,407,412,415,423,431,439,444,447,448,455,463,471,476,479,487,
> 495,496,503,508,511,519,527,535,540,543,551,559,567,572,575,583,591,599,604,
> 607,615,623,624,631,636,639,647,655,663,668,671,679,687,695,700,703,711,719,
> 727,732 etc...

   and then made some comments about the differences of these numbers.

   I didn't follow them in detail, but it seems likely that they
follow from Legendre's famous theorem that the natural numbers
that can't be expressed as the sum of three squares are precisely
those of the form  4^a.(8b+7).

         John Conway