A347827 Number of ways to write n as w^4 + x^4 + (y^2 + 23*z^2)/16, where w is zero or a power of two (including 2^0 = 1), and x,y,z are nonnegative integers.

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

Offset: 0

Author

Zhi-Wei Sun, Jan 23 2022

Keywords

• nonn

Comments

23-Conjecture: a(n) > 0 for all n = 0,1,2,....

This is stronger than the conjecture in A347824, and it has been verified for n up to 3*10^6. See also a similar conjecture in A347562.

It seems that a(n) = 1 only for n = 0, 15, 77, 231, 291, 437, 471, 1161, 1402, 4692, 7107, 9727.

Examples

			a(231) = 1 with 231 = 0^4 + 3^4 + (10^2 + 23*10^2)/16.
a(437) = 1 with 437 = 3^4 + 4^4 + (40^2 + 23*0^2)/16.
a(1402) = 1 with 1402 = 2^4 + 5^4 + (3^2 + 23*23^2)/16.
a(9727) = 1 with 9727 = 0^4 + 6^4 + (367^2 + 23*3^2)/16.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Crossrefs

Cf. A000079, A000290, A000583, A347562, A347824.

Programs

Mathematica

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