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%I #27 Jul 07 2023 18:47:01
%S 1,3,4,4,4,3,2,2,2,3,4,4,4,2,1,1,2,5,5,5,4,1,1,1,2,4,5,6,6,3,3,2,3,7,
%T 6,4,4,5,4,3,3,4,5,4,5,4,3,2,2,8,5,3,6,4,3,2,2,5,4,4,5,2,1,2,5,9,7,5,
%U 7,4,3,1,2,4,3,5,5,2,1,3,3,8,9,8,8,5,2,1,2,5,6,7,7,3,2,2,4,7,7,2,7
%N Number of ordered ways to write n as w^2 + x^3 + y^4 + 2*z^4, where w, x, y and z are nonnegative integers.
%C Conjecture: a(n) > 0 for all n >= 0. Also, a(n) = 1 only for the following 41 values of n: 0, 14, 15, 21, 22, 23, 62, 71, 78, 87, 136, 216, 405, 437, 448, 477, 535, 583, 591, 623, 671, 696, 885, 950, 1046, 1135, 1206, 1208, 1248, 1317, 2288, 2383, 2543, 3167, 3717, 3974, 6847, 7918, 8328, 9096, 21935.
%C We have verified that a(n) > 0 for all n = 0..10^7.
%C Conjecture verified up to 10^11. - _Mauro Fiorentini_, Jul 07 2023
%C We also conjecture that if f(w,x,y,z) is one of the 8 polynomials 2w^2+x^3+4y^3+z^4, w^2+x^3+2y^3+c*z^3 (c = 3,4,5,6) and w^2+x^3+2y^3+d*z^4 (d = 1,3,6) then each n = 0,1,2,... can be written as f(w,x,y,z) with w,x,y,z nonnegative integers. - _Zhi-Wei Sun_, Dec 30 2017
%C Conjecture verified up to 10^11 for all 8 polynomials. - _Mauro Fiorentini_, Jul 07 2023
%H Zhi-Wei Sun, <a href="/A262827/b262827.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. (See Remark 1.1.)
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. (See Theorem 1.1 and Conjecture 1.1.)
%e a(14) = 1 since 14 = 2^2 + 2^3 + 0^4 + 2*1^4.
%e a(87) = 1 since 87 = 2^2 + 0^3 + 3^4 + 2*1^4.
%e a(216) = 1 since 216 = 0^2 + 6^3 + 0^4 + 2*0^4.
%e a(405) = 1 since 405 = 18^2 + 0^3 + 3^4 + 2*0^4.
%e a(1248) = 1 since 1248 = 31^2 + 5^3 + 0^4 + 2*3^4.
%e a(1317) = 1 since 1317 = 23^2 + 1^3 + 5^4 + 2*3^4.
%e a(2288) = 1 since 2288 = 44^2 + 4^3 + 4^4 +2*2^4.
%e a(2383) = 1 since 2383 = 1462 + 9^3 + 6^4 + 2*3^4.
%e a(2543) = 1 since 2543 = 50^2 + 3^3 + 2^4 + 2*0^4.
%e a(3167) = 1 since 3167 = 54^2 + 2^3 + 3^4 + 2*3^4.
%e a(3717) = 1 since 3717 = 18^2 + 15^3 + 2^4 + 2*1^4.
%e a(3974) = 1 since 3974 = 39^2 + 13^3 + 4^4 + 2*0^4.
%e a(6847) = 1 since 6847 = 52^2 + 15^3 + 4^4 + 2*4^4.
%e a(7918) = 1 since 7918 = 46^2 + 10^3 + 0^4 + 2*7^4.
%e a(8328) = 1 since 8328 = 42^2 + 1^3 + 9^4 + 2*1^4.
%e a(9096) = 1 since 9096 = 44^2 + 18^3 + 6^4 + 2*2^4.
%e a(21935) = 1 since 21935 = 66^2 + 26^3 + 1^4 + 2*1^4.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t Do[r=0;Do[If[SQ[n-x^3-y^4-2*z^4],r=r+1],{x,0,n^(1/3)},{y,0,(n-x^3)^(1/4)},{z,0,((n-x^3-y^4)/2)^(1/4)}];Print[n," ",r];Continue,{n,0,100}]
%Y Cf. A000290, A000578, A000583, A262813, A262815, A262816, A262824.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Oct 03 2015
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