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

2, 5, 8, 10, 13, 20, 25, 32, 37, 40, 52, 58, 80, 85, 100, 128, 130, 148, 160, 208, 232, 320, 340, 400, 512, 520, 592, 640, 832, 928, 1280, 1360, 1600, 2048, 2080, 2368, 2560, 3328, 3712, 5120, 5440, 6400, 8192, 8320, 9472, 10240, 13312, 14848, 20480, 21760

Offset: 1

Author

Michel Marcus, Nov 16 2012

Keywords

• nonn

Comments

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.

Subsequence of A000549. - Chai Wah Wu, Feb 05 2016

Links

• Donovan Johnson, Table of n, a(n) for n = 1..120 (terms <= 10^9)
• P. K. J. Draxl, Sommes de deux carrés qui ne sont pas sommes de trois carrés., Mémoires de la SMF, tome 37 (1974), p. 53-53.

Crossrefs

Cf. A000404, A000549, A051952.

Programs

Python

limit = 21760
squares_lst = [i*i for i in range(1, int(limit**0.5)+2) if i*i <= limit]
squares_set = set(squares_lst)
def sum2squares(n):
  for s in squares_lst:
    if n - s in squares_set: return True
    if n - s < 0: return False
alst = []
for m in range(2, limit+1):
  if sum2squares(m):
    sum3 = False
    for s in squares_lst:
      if sum2squares(m - s): sum3 = True; break
      if m - s < 0: break
    if not sum3: alst.append(m)
print(alst) # Michael S. Branicky, Feb 05 2021

Formula

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