This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A272979 #34 Jul 14 2016 00:17:35
%S A272979 1,1,1,2,3,2,3,3,3,4,2,3,4,3,1,3,4,1,3,3,2,3,4,2,3,5,3,4,4,3,4,4,4,4,
%T A272979 4,2,7,5,2,4,6,4,3,4,3,3,4,3,4,2,3,6,3,3,5,5,2,7,5,1,5,6,3,1,6,2,5,5,
%U A272979 5,4,5
%N A272979 Number of ways to write n as x^2 + 2*y^2 + 3*z^3 + 4*w^4 with x,y,z,w nonnegative integers.
%C A272979 Conjecture: For positive integers a,b,c,d, any natural number can be written as a*x^2 + b*y^2 + c*z^3 + d*w^4 with x,y,z,w nonnegative integers, if and only if (a,b,c,d) is among the following 49 quadruples: (1,2,1,1), (1,3,1,1), (1,6,1,1), (2,3,1,1), (2,4,1,1), (1,1,2,1), (1,4,2,1), (1,2,3,1), (1,2,4,1), (1,2,12,1), (1,1,1,2), (1,2,1,2), (1,3,1,2), (1,4,1,2), (1,5,1,2), (1,11,1,2), (1,12,1,2), (2,4,1,2), (3,5,1,2), (1,1,4,2), (1,1,1,3), (1,2,1,3), (1,3,1,3), (1,2,4,3), (1,2,1,4), (1,3,1,4), (2,3,1,4), (1,1,2,4), (1,2,2,4), (1,8,2,4), (1,2,3,4), (1,1,1,5), (1,2,1,5), (2,3,1,5), (2,4,1,5), (1,3,2,5), (1,1,1,6), (1,3,1,6), (1,1,2,6), (1,2,1,8), (1,2,4,8), (1,2,1,10), (1,1,2,10), (1,2,1,11), (2,4,1,11), (1,2,1,12), (1,1,2,13), (1,2,1,14),(1,2,1,15).
%C A272979 See also A262824, A262827, A262857 and A273917 for similar conjectures.
%H A272979 Zhi-Wei Sun, <a href="/A272979/b272979.txt">Table of n, a(n) for n = 0..10000</a>
%e A272979 a(0) = 1 since 0 = 0^2 + 2*0^2 + 3*0^3 + 4*0^4.
%e A272979 a(1) = 1 since 1 = 1^2 + 2*0^2 + 3*0^3 + 4*0^4.
%e A272979 a(2) = 1 since 2 = 0^2 + 2*1^2 + 3*0^3 + 4*0^4.
%e A272979 a(14) = 1 since 14 = 3^2 + 2*1^2 + 3*1^3 + 4*0^4.
%e A272979 a(17) = 1 since 17 = 3^2 + 2*2^2 + 3*0^3 + 4*0^4.
%e A272979 a(59) = 1 since 59 = 3^2 + 2*5^2 + 3*0^3 + 4*0^4.
%e A272979 a(63) = 1 since 63 = 3^2 + 2*5^2 + 3*0^2 + 4*1^4.
%e A272979 a(287) = 1 since 287 = 11^2 + 2*9^2 + 3*0^2 + 4*1^4.
%t A272979 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A272979 Do[r=0;Do[If[SQ[n-4w^4-3z^3-2y^2],r=r+1],{w,0,(n/4)^(1/4)},{z,0,((n-4w^4)/3)^(1/3)},{y,0,((n-4w^4-3z^3)/2)^(1/2)}];Print[n," ",r];Continue,{n,0,100}]
%Y A272979 Cf. A000290, A000578, A000583, A262824, A262827, A262857, A270969, A273429, A273915, A273917.
%K A272979 nonn
%O A272979 0,4
%A A272979 _Zhi-Wei Sun_, Jul 13 2016