A346643 Number of ways to write n as w^2 + 2*x^2 + 3*y^4 + 4*z^4, where w,x,y,z are nonnegative integers.

1, 1, 1, 2, 3, 2, 3, 3, 3, 4, 2, 3, 4, 3, 1, 3, 4, 1, 3, 3, 2, 3, 4, 2, 2, 4, 2, 3, 2, 2, 2, 3, 2, 2, 3, 1, 5, 3, 2, 3, 4, 3, 1, 3, 2, 2, 1, 2, 4, 2, 3, 5, 4, 3, 7, 4, 3, 7, 5, 2, 4, 6, 1, 2, 6, 2, 6, 5, 5, 4, 8, 5, 5, 7, 2, 8, 8, 2, 2, 7, 4, 6, 5, 4, 7, 8, 7, 1, 7, 6, 3, 5, 4, 3, 2, 2, 5, 4, 3, 7, 8

Offset: 0

Author

Zhi-Wei Sun, Jan 24 2022

Keywords

• nonn

Comments

1-2-3-4 Conjecture: a(n) > 0 except for n = 158.

This has been verified for n up to 10^8.

It seems that a(n) = 1 only for n = 0, 1, 2, 14, 17, 35, 42, 46, 62, 87, 119, 122, 168, 189, 206, 234, 237, 302, 317, 398, 545, 1037, 1437, 4254.

See also A347865 and A350857 for similar conjectures.

Examples

			a(46) = 1 with 46 = 5^2 + 2*3^2 + 3*1^4 + 4*0^4.
a(119) = 1 with 119 = 7^2 + 2*3^2 + 3*2^4 + 4*1^4.
a(398) = 1 with 398 = 13^2 + 2*9^2 + 3*1^4 + 4*2^4.
a(545) = 1 with 545 = 19^2 + 2*6^2 + 3*2^4 + 4*2^4.
a(1037) = 1 with 1037 = 31^2 + 2*6^2 + 3*0^4 + 4*1^4.
a(1437) = 1 with 1437 = 9^2 + 2*26^2 + 3*0^4 + 4*1^4.
a(4254) = 1 with 4254 = 45^2 + 2*31^2 + 3*3^4 + 4*2^4.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
• Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Crossrefs

Cf. A000290, A000583, A347865, A350857.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-4x^4-3y^4-2z^2],r=r+1],{x,0,(n/4)^(1/4)},{y,0,((n-4x^4)/3)^(1/4)},{z,0,Sqrt[(n-4x^4-3y^4)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]