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

%I #28 Feb 06 2021 00:05:15

%S 2,5,8,10,13,20,25,32,37,40,52,58,80,85,100,128,130,148,160,208,232,

%T 320,340,400,512,520,592,640,832,928,1280,1360,1600,2048,2080,2368,

%U 2560,3328,3712,5120,5440,6400,8192,8320,9472,10240,13312,14848,20480,21760

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

%C 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 Subsequence of A000549. - _Chai Wah Wu_, Feb 05 2016

%H Donovan Johnson, <a href="/A219222/b219222.txt">Table of n, a(n) for n = 1..120</a> (terms <= 10^9)

%H 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 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 (Python)

%o limit = 21760

%o squares_lst = [i*i for i in range(1, int(limit**0.5)+2) if i*i <= limit]

%o squares_set = set(squares_lst)

%o def sum2squares(n):

%o for s in squares_lst:

%o if n - s in squares_set: return True

%o if n - s < 0: return False

%o alst = []

%o for m in range(2, limit+1):

%o if sum2squares(m):

%o sum3 = False

%o for s in squares_lst:

%o if sum2squares(m - s): sum3 = True; break

%o if m - s < 0: break

%o if not sum3: alst.append(m)

%o print(alst) # _Michael S. Branicky_, Feb 05 2021

%Y Cf. A000404, A000549, A051952.

%K nonn

%O 1,1

%A _Michel Marcus_, Nov 16 2012