A347562 Number of ways to write n as 16^w + x^2 + (y^2 + 23*z^4)/324, where w,x,y,z are nonnegative integers.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 23 2022

Keywords

• nonn

Comments

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

This has been verified for n up to 10^6.

It seems that a(n) = 1 only for n = 1, 4, 8, 12, 15, 40, 55, 70, 76, 85, 92, 104, 135, 156, 177, 192, 204, 207, 231, 279, 300, 447, 567, 1427, 1887, 4371.

See also A347827 for a similar conjecture.

Examples

			a(15) = 1 with 15 = 16^0 + 1^2 + (62^2 + 23*2^4)/324.
a(156) = 1 with 156 = 16^1 + 6^2 + (139^2 + 23*5^4)/324.
a(300) = 1 with 300 = 16^2 + 6^2 + (27^2 + 23*3^4)/324.
a(1427) = 1 with 1427 = 16^1 + 35^2 + (71^2 + 23*7^4)/324.
a(1887) = 1 with 1887 = 16^1 + 15^2 + (729^2 + 23*3^4)/324.
a(4371) = 1 with 4371 = 16^1 + 63^2 + (351^2 + 23*3^4)/324.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Crossrefs

Cf. A000290, A000583, A001025, A347824, A347827, A350860.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[324(n-16^w-x^2)-23y^4],r=r+1],{w,0,Log[16,n]},{x,0,Sqrt[n-16^w]},
{y,0,(324(n-16^w-x^2)/23)^(1/4)}];tab=Append[tab,r],{n,1,100}];Print[tab]