Questions on sums of squares/PARI

[franktaw at netscape.net]

Can this be extended to give an actual upper bound on the elements of
A138555, so that we can perhaps verify that the given list is complete?

(It looks to me like this should be possible.)

Franklin T. Adams-Watters

-----Original Message-----
From: Max Alekseyev <maxale at gmail.com>

Rustem Aidagulov has outlined the following proof of finiteness of 
A138555:

1) Reformulate the definition of A138554 as follows:
(*)    A138554(n) = min (k + A138554(n-k^2)),
where k goes over 1..[sqrt(n)].

2) Prove by induction on n that
[sqrt(n)] =< A138554(n) < [sqrt(n)] + 2*n^(1/4) + 1.6

3) The above inequality implies that if k_1^2 + ... + k_s^2 = n and
A138554(n) = k_1 + ... + k_s where k_1 <= ... <= k_s, then k_s =
[sqrt(n)] or [sqrt(n)] - 1.

4) By direct comparison of computations of (*) for k = [sqrt(n)] and k
= [sqrt(n)] - 1, using the bounds 2), derive that the latter value can
be smaller than the former one only for finitely many n. This prove
the finiteness of A138555.

I did not really check all these arguments,
so it would be nice if somebody check them up along with all missing 
details.

Regards,
Max