A302985 Number of ordered pairs (x, y) of nonnegative integers such that n - 2^x - 3*2^y has the form u^2 + 2*v^2 with u and v integers.

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

Offset: 1

Author

Zhi-Wei Sun, Apr 16 2018

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 3.

This is equivalent to the author's conjecture in A302983. We have verified a(n) > 0 for all n = 4...6*10^9.

See also A302982 and A302984 for similar conjectures.

Examples

			a(4) = 1 with 4 - 2^0 - 3*2^0 = 0^2 + 2*0^2.
a(5) = 2 with 5 - 2^0 - 3*2^0 = 1^2 + 2*0^2 and 5 - 2^1 - 3*2^0 = 0^2 + 2*0^2.
a(6) = 2 with 6 - 2^0 - 3*2^0 = 0^2 + 2*1^2 and 6 - 2^1 - 3*2^0 = 1^2 + 2*0^2.
		

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, A302981, A302982, A302983, A302984.

Programs

Mathematica

  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-3*2^k-2^j],r=r+1],{k,0,Log[2,n/3]},{j,0,If[3*2^k==n,-1,Log[2,n-3*2^k]]}];tab=Append[tab,r],{n,1,60}];Print[tab]