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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

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

PROG

(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

CROSSREFS

Cf. A000404, A000549, A051952.

Sequence in context: A070216 A100829 A030713 * A263831 A189457 A189362

Adjacent sequences:  A219219 A219220 A219221 * A219223 A219224 A219225

KEYWORD

nonn

AUTHOR

Michel Marcus, Nov 16 2012

STATUS

approved

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Last modified May 22 20:08 EDT 2022. Contains 353957 sequences. (Running on oeis4.)