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.

1, 5, 29, 65

Offset

1,2

Comments

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.

Links

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

Mathematica

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]

Crossrefs

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

Sequence in context: A103094 A341083 A341085 * A293174 A108928 A097812

Adjacent sequences: A301855 A301856 A301857 * A301859 A301860 A301861

Keyword

nonn

Author

Zhi-Wei Sun, Mar 27 2018

Status

approved