A000549 -id:A000549 - cp's OEIS frontend

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

Michel Marcus, Nov 16 2012

nonn

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

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.

Cf. A000404, A000549, A051952.

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

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

A301858 Positive integers which can be written as the sum of two squares but cannot be written as x^2 + y^2 + 2*z^2 with x and y integers and z a nonzero integer.

Original entry on oeis.org

1, 5, 29, 65

Offset: 1

Zhi-Wei Sun, Mar 27 2018

nonn

The sequence has no term in the interval [66, 10^6].
Conjecture 1: The sequence only has the four terms 1, 5, 29 and 65.
Conjecture 2: For any integer n > 1 which is neither 17 nor a power of 2, if n = u^2 + 2*v^2 for some integers u and v, then n = x^2 + 2*y^2 + 3*z^2 for some integers x,y,z with z nonzero.
Conjecture 3: For any positive integer n not of the form 4^k*m (k = 0,1,2,... and m = 1, 7, 13), if n = u^2 + 3*v^2 for some integers u and v, then n = x^2 + 2*y^2 + 3*z^2 for some integers x,y,z with y nonzero.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4

Cf. A000290, A000549, A001481, A002479, A301471.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[If[QQ[m]==False,Goto[aa]];Do[If[SQ[m-2x^2-y^2],Goto[aa]],{x,1,Sqrt[m/2]},{y,0,Sqrt[(m-2x^2)/2]}];tab=Append[tab,m];Label[aa],{m,1,1000}];Print[tab]

cp's OEIS Frontend

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

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