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A219222 Numbers that can be expressed as the sum of 2 positive squares but not as the sum of 3 positive squares.

%I A219222 #28 Feb 06 2021 00:05:15
%S A219222 2,5,8,10,13,20,25,32,37,40,52,58,80,85,100,128,130,148,160,208,232,
%T A219222 320,340,400,512,520,592,640,832,928,1280,1360,1600,2048,2080,2368,
%U A219222 2560,3328,3712,5120,5440,6400,8192,8320,9472,10240,13312,14848,20480,21760
%N A219222 Numbers that can be expressed as the sum of 2 positive squares but not as the sum of 3 positive squares.
%C A219222 Among these numbers a(n), some of them are not divisible by 4: 2, 5, 10, 13, 25, 37, 58, 85, 130. All members of the sequence can be expressed as a(n) = 4^k*a0, with a0 taken in the set described above, that is A051952 except 1.
%C A219222 Subsequence of A000549. - _Chai Wah Wu_, Feb 05 2016
%H A219222 Donovan Johnson, <a href="/A219222/b219222.txt">Table of n, a(n) for n = 1..120</a> (terms <= 10^9)
%H A219222 P. K. J. Draxl, <a href="http://www.numdam.org/item?id=MSMF_1974__37__53_0">Sommes de deux carrés qui ne sont pas sommes de trois carrés.</a>, Mémoires de la SMF, tome 37 (1974), p. 53-53.
%F A219222 Empirical g.f.: -x*(2*x^16 +28*x^15 +20*x^14 +33*x^13 +40*x^12 +26*x^11 +32*x^10 +32*x^9 +37*x^8 +32*x^7 +25*x^6 +20*x^5 +13*x^4 +10*x^3 +8*x^2 +5*x +2) / (4*x^9 -1). - _Colin Barker_, Sep 23 2014
%o A219222 (Python)
%o A219222 limit = 21760
%o A219222 squares_lst = [i*i for i in range(1, int(limit**0.5)+2) if i*i <= limit]
%o A219222 squares_set = set(squares_lst)
%o A219222 def sum2squares(n):
%o A219222   for s in squares_lst:
%o A219222     if n - s in squares_set: return True
%o A219222     if n - s < 0: return False
%o A219222 alst = []
%o A219222 for m in range(2, limit+1):
%o A219222   if sum2squares(m):
%o A219222     sum3 = False
%o A219222     for s in squares_lst:
%o A219222       if sum2squares(m - s): sum3 = True; break
%o A219222       if m - s < 0: break
%o A219222     if not sum3: alst.append(m)
%o A219222 print(alst) # _Michael S. Branicky_, Feb 05 2021
%Y A219222 Cf. A000404, A000549, A051952.
%K A219222 nonn
%O A219222 1,1
%A A219222 _Michel Marcus_, Nov 16 2012