A349942 - OEIS

%I #12 Dec 06 2021 08:28:37

%S 1,4,6,4,3,5,4,1,1,4,8,7,2,4,6,2,4,12,13,6,7,9,4,1,2,11,19,11,2,10,10,

%T 2,6,12,12,9,11,9,8,4,3,16,18,7,1,13,10,1,4,7,17,15,11,11,10,2,4,12,

%U 11,9,4,13,12,5,3,15,25,10,10,12,8,3,4,9,17,17,4,14,16,3,5,20,20,14,13,12,14,4,3,12,30,22,3,12,13,4,4,16,24,20,11

%N Number of ways to write n as a^4 + b^2 + (c^4 + d^2)/25 with a,b,c,d nonnegative integers.

%C Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 7, 8, 23, 44, 47).

%C We have verified this for n up to 3*10^5.

%C As m/n = (m*n^3)/n^4 for any nonnegative integers m and n > 0, the conjecture implies that each nonnegative rational number can be written as x^4 + 25*y^4 + z^2 + w^2 with x,y,z,w rational numbers.

%C See also A349943 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A349942/b349942.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167--190.

%H Zhi-Wei Sun, <a href="http://hitpress.hit.edu.cn/2021/1015/c12593a261001/page.htm">New Conjectures in Number Theory and Combinatorics</a> (in Chinese), Harbin Institute of Technology Press, 2021.

%e a(0) = 1 with 0 = 0^4 + 0^2 + (0^4 + 0^2)/25.

%e a(7) = 1 with 7 = 1^4 + 2^2 + (1^4 + 7^2)/25.

%e a(8) = 1 with 8 = 0^4 + 2^2 + (0^4 + 10^2)/25.

%e a(23) = 1 with 23 = 1^4 + 3^2 + (1^4 + 18^2)/25.

%e a(44) = 1 with 44 = 1^4 + 3^2 + (5^4 + 15^2)/25.

%e a(47) = 1 with 47 = 1^4 + 6^2 + (3^4 + 13^2)/25.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t tab={};Do[r=0;Do[If[SQ[25(n-x^4-y^2)-z^4],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,(25(n-x^4-y^2))^(1/4)}];tab=Append[tab,r],{n,0,100}];Print[tab]

%Y Cf. A000290, A000583, A349943.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Dec 05 2021