A302981 Number of ways to write n as x^2 + 2*y^2 + 2^z + 2^w, where x,y,z,w are nonnegative integers with z <= w.

0, 1, 2, 3, 4, 5, 4, 5, 5, 6, 7, 8, 7, 8, 6, 6, 7, 9, 8, 11, 12, 9, 7, 10, 8, 11, 11, 11, 10, 9, 6, 8, 10, 11, 14, 16, 12, 12, 11, 12, 13, 17, 13, 13, 13, 10, 7, 11, 12, 13, 15, 15, 14, 14, 8, 15, 14, 13, 15, 16

Offset: 1

Author

Zhi-Wei Sun, Apr 16 2018

Keywords

• nonn

Comments

Clearly, a(2*n) > 0 if a(n) > 0. We note that 52603423 is the first value of n > 1 with a(n) = 0.

See also A302982 and A302983 for related things.

Examples

			a(2) = 1 with 2 = 0^2 + 2*0^2 + 2^0 + 2^0.
a(3) = 2 with 3 = 1^2 + 2*0^2 + 2^0 + 2^0 = 0^2 + 2*0^2 + 2^0 + 2^1.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
• Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Crossrefs

Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302982, A302983.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[MemberQ[{5,7},Mod[Part[Part[f[n],i],1],8]]&&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]);
tab={};Do[r=0;Do[If[QQ[n-2^k-2^j],Do[If[SQ[n-2^k-2^j-2x^2],r=r+1],{x,0,Sqrt[(n-2^k-2^j)/2]}]],{k,0,Log[2,n]-1},{j,k,Log[2,Max[1,n-2^k]]}];tab=Append[tab,r],{n,1,60}];Print[tab]