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A271381 Number of ordered ways to write n as u^2 + v^2 + x^3 + y^3, where u, v, x, y are nonnegative integers with 2 | u*v, u <= v and x <= y. 1

%I

%S 1,2,2,1,1,2,2,1,2,4,3,1,1,3,2,1,3,5,3,1,2,3,2,0,2,5,3,3,3,4,1,2,4,5,

%T 3,2,5,4,3,2,4,6,2,3,4,5,2,2,4,3,2,2,5,6,4,3,2,3,2,2,4,4,3,3,5,7,4,5,

%U 5,6,4,2,6,9,6,2,4,5,1,3,8

%N Number of ordered ways to write n as u^2 + v^2 + x^3 + y^3, where u, v, x, y are nonnegative integers with 2 | u*v, u <= v and x <= y.

%C Conjecture: (i) a(n) > 0 except for n = 23, and a(n) = 1 only for n = 0, 3, 4, 7, 11, 12, 15, 19, 23, 30, 78, 203, 219.

%C (ii) Any natural number can be written as the sum of two squares and three fourth powers.

%H Zhi-Wei Sun, <a href="/A271381/b271381.txt">Table of n, a(n) for n = 0..10000</a>

%e a(3) = 1 since 3 = 0^2 + 1^2 + 1^3 + 1^3 with 0 even.

%e a(4) = 1 since 4 = 0^2 + 2^2 + 0^3 + 0^3 with 0 and 2 even.

%e a(7) = 1 since 7 = 1^2 + 2^2 + 1^3 + 1^3 with 2 even.

%e a(11) = 1 since 11 = 0^2 + 3^2 + 1^3 + 1^3 with 0 even.

%e a(12) = 1 since 12 = 0^2 + 2^2 + 0^3 + 2^3 with 0 and 2 even.

%e a(15) = 1 since 15 = 2^2 + 3^2 + 1^3 + 1^3 with 2 even.

%e a(19) = 1 since 19 = 1^2 + 4^2 + 1^3 + 1^3 with 4 even.

%e a(30) = 1 since 30 = 2^2 + 5^2 + 0^3 + 1^3 with 2 even.

%e a(78) = 1 since 78 = 2^2 + 3^2 + 1^3 + 4^3 with 2 even.

%e a(203) = 1 since 203 = 7^2 + 10^2 + 3^3 + 3^3 with 10 even.

%e a(219) = 1 since 219 = 8^2 + 8^2 + 3^3 + 4^3 with 8 even.

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

%t Do[r=0;Do[If[SQ[n-x^3-y^3-u^2]&&(Mod[u*Sqrt[n-x^3-y^3-u^2],2]==0),r=r+1],{x,0,(n/2)^(1/3)},{y,x,(n-x^3)^(1/3)},{u,0,((n-x^3-y^3)/2)^(1/2)}];Print[n," ",r];Continue,{n,0,80}]

%Y Cf. A000290, A000578, A000583, A262813, A262824, A262827, A262857, A262880, A270488, A270559, A270969, A271325.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Apr 06 2016

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Last modified June 15 20:50 EDT 2019. Contains 324145 sequences. (Running on oeis4.)