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The number in the title is 5/6, which is the same as the asymptotic density of the set. The best possible upper and lower bounds (of the form 5x/6 + A + B log x) are obtained for the number of positive integers less than x which are a sum of three squares. For x > 0 let S(x) and N(x) be the numbers of positive integers not greater than x which are and are not a sum of three squares of integers, respectively. It is well known that N(x) is the number of numbers <x which are of the form 4a(8b + 7) where a and b are nonnegative integers. Since there are [(1 + x)/8] integers k < x with k _ 7 (mod 8), it is easy to see that

The Schnirelmann density of the sums of three squares