A349945 - OEIS

%I #34 Dec 09 2021 08:15:28

%S 1,3,4,3,3,6,7,3,1,4,5,4,2,3,8,5,3,9,10,6,7,11,10,3,2,6,8,9,3,9,16,5,

%T 4,11,9,7,9,9,12,7,2,8,11,7,2,11,14,4,3,10,10,9,8,9,21,9,3,9,5,7,4,10,

%U 17,8,3,15,15,9,9,16,20,5,3,5,7,11,3,11,18,4,6,22,18,11,14,15,19,10,2,9,16,10,3,9,16,11,7,19,16,13,12

%N Number of ways to write n as a^4 + b^2 + (c^4 + d^2)/5 with a,b,c,d nonnegative integers.

%C Conjecture 1: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 2^(4*k+3) (k = 0,1,2,...).

%C This has been verified for all n <= 10^5.

%C Conjecture 2: Each n = 0,1,2,... can be written as a*x^4 + b*y^2 + (c*z^4 + w^2)/5 with x,y,z,w nonnegative integers, provided that (a,b,c) is among the four triples (1,2,4), (2,1,1), (6,1,1), (6,1,6).

%C See also A349942 for a similar conjecture.

%C Via a computer search, we have found many tuples (a,b,c,d,m) of positive integers (such as (1,1,4,2,3), (4,1,1,2,3) and (1,1,19,1,4900)) for which we guess that each n = 0,1,2,... can be written as a*x^4 + b*y^2 + (c*z^4 + d*w^2)/m with x,y,z,w nonnegative integers.

%H Zhi-Wei Sun, <a href="/A349945/b349945.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(8) = 1 with 8 = 0^4 + 2^2 + (2^4 + 2^2)/5.

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

%t tab={};Do[r=0;Do[If[SQ[5(n-x^4-y^2)-z^4],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,(5(n-x^4-y^2))^(1/4)}];tab=Append[tab,r],{n,0,100}];Print[tab]

%Y Cf. A000290, A000583, A349942, A349956, A349957.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Dec 06 2021