A347865 Number of ways to write n as w^2 + 2*x^2 + y^4 + 3*z^4, where w,x,y,z are nonnegative integers.

1, 2, 2, 3, 4, 3, 3, 3, 2, 4, 3, 2, 5, 3, 1, 2, 3, 3, 4, 6, 5, 4, 6, 3, 2, 6, 2, 5, 7, 1, 3, 3, 2, 4, 5, 4, 6, 7, 4, 3, 3, 4, 2, 4, 4, 2, 3, 2, 4, 6, 5, 7, 10, 4, 7, 7, 1, 9, 6, 3, 7, 3, 2, 2, 4, 5, 7, 11, 6, 4, 9, 3, 5, 11, 2, 7, 10, 2, 2, 2, 4, 8, 12, 7, 9, 10, 7, 6, 5, 7, 6, 7, 8, 5, 1, 2, 4, 10, 7, 11, 15

Offset: 0

Author

Zhi-Wei Sun, Jan 24 2022

Keywords

• nonn

Comments

Conjecture 1: a(n) > 0 except for n = 744.

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

It seems that a(n) = 1 only for n = 0, 14, 29, 56, 94, 110, 158, 159, 224, 239, 296, 464, 589, 1214, 1454, 1709.

Conjecture 2: For any positive odd integer a, all sufficiently large integers can be written as a*w^4 + 2*x^4 + (2*y)^2 + z^2 with w,x,y,z integers. If M(a) denotes the largest integer not of the form a*w^4 + 2*x^4 + (2*y)^2 + z^2 (with w,x,y,z integers), then M(1) = 255, M(3) = 303, M(5) = 497, M(7) = 3182, M(9) = 4748, M(11) = 5662, M(13) = 5982, M(15) = 10526, M(17) = 4028 and M(19) = 11934.

Conjecture 3: Let E(a,b,c) be the set of nonnegative integers not of the form w^2 + a*x^2 + b*y^4 + c*z^4 with w,x,y,z integers. Then E(1,2,4) = {135, 190, 510}, E(1,2,5) = {35, 254, 334}, E(2,1,4) = {190, 270, 590} and E(2,3,7) = {94, 490, 983} and E(3,1,2) = {56, 168, 378}.

See also A346643 and A350857 for similar conjectures.

Examples

			a(14) = 1 with 14 = 3^2 + 2*1^2 + 0^4 + 3*1^4.
a(158) = 1 with 158 = 11^2 + 2*3^2 + 2^4 + 3*1^4.
a(589) = 1 with 589 = 14^2 + 2*14^2 + 1^4 + 3*0^4.
a(1214) = 1 with 1214 = 27^2 + 2*11^2 + 0^4 + 3*3^4.
a(1454) = 1 with 1454 = 27^2 + 2*19^2 + 0^4 + 3*1^4.
a(1709) = 1 with 1709 = 29^2 + 2*0^2 + 5^4 + 3*3^4.
		

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, A346643, A350857.

Programs

Mathematica

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