cp's OEIS Frontend

A275300 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^3 with x + y + z a square, where x,y,z are integers with x >= |y| <= |z|, and w is a nonnegative integer.

%I A275300 #9 Jul 23 2016 09:23:15
%S A275300 1,3,3,3,2,1,5,4,3,5,4,5,1,2,9,4,4,4,7,6,1,2,6,1,7,7,8,6,3,5,7,1,7,11,
%T A275300 11,9,4,5,6,4,3,15,10,8,2,7,9,1,4,9,5,12,5,11,10,3,8,5,3,8,7,10,10,2,
%U A275300 4,11,9,8,6,10,13,1,7,10,8,8,2,10,14,3,10
%N A275300 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^3 with x + y + z a square, where x,y,z are integers with x >= |y| <= |z|, and w is a nonnegative integer.
%C A275300 Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 5, 12, 20, 23, 31, 47, 71, 103, 148, 164.
%C A275300 The author proved in arXiv:1604.06723 that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x + y + z a square, where x,y,z,w are integers.
%C A275300 See also A275297, A275298, A275299 and A272620 for similar conjectures.
%H A275300 Zhi-Wei Sun, <a href="/A275300/b275300.txt">Table of n, a(n) for n = 0..10000</a>
%H A275300 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.
%e A275300 a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 0^3 with 0 + 0 + 0 = 0^2 and 0 = 0 = 0.
%e A275300 a(5) = 1 since 5 = 2^2 + 0^2 + (-1)^2 + 0^3 with 2 + 0 + (-1) = 1^2 and 2 > 0 < |-1|.
%e A275300 a(12) = 1 since 12 = 3^2 + (-1)^2 + (-1)^2 + 1^3 with 3 + (-1) + (-1) = 1^2 and 3 > |-1| = |-1|.
%e A275300 a(20) = 1 since 20 = 3^2 + 1^2 + (-3)^2 + 1^3 with 3 + 1 + (-3) = 1^2 and 3 > 1 < |-3|.
%e A275300 a(23) = 1 since 23 = 3^2 + (-2)^2 + 3^2 + 1^3 with 3 + (-2) + 3 = 2^2 and 3 > |-2| < 3.
%e A275300 a(31) = 1 since 31 = 5^2 + 1^2 + (-2)^2 + 1^3 with 5 + 1 + (-2) = 2^2 and 5 > 1 < |-2|.
%e A275300 a(47) = 1 since 47 = 6^2 + 1^2 + (-3)^2 + 1^3 with 6 + 1 + (-3) = 2^2 and 6 > 1 < |-3|.
%e A275300 a(71) = 1 since 71 = 6^2 + 3^2 + (-5)^2 + 1^3 with 6 + 3 + (-5) = 2^2 and 6 > 3 < |-5|.
%e A275300 a(103) = 1 since 103 = 7^2 + 2^2 + 7^2 + 1^3 with 7 + 2 + 7 = 4^2 and 7 > 2 < 7.
%e A275300 a(148) = 1 since 148 = 9^2 + (-2)^2 + (-6)^2 + 3^3 with 9 + (-2) + (-6) = 1^2 and 9 > |-2| < |-6|.
%e A275300 a(164) = 1 since 164 = 9^2 + 1^2 + (-9)^2 + 1^3 with 9 + 1 + (-9) = 1^2 and 9 > 1 < |-9|.
%t A275300 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A275300 Do[r=0;Do[If[SQ[n-w^3-y^2-z^2]&&SQ[Sqrt[n-w^3-y^2-z^2]+(-1)^i*y+(-1)^j*z],r=r+1],{w,0,n^(1/3)},{y,0,Sqrt[(n-w^3)/3]},{i,0,Min[1,y]},{z,y,Sqrt[n-w^3-2y^2]},{j,0,Min[1,z]}];Print[n," ",r];Continue,{n,0,80}]
%Y A275300 Cf. A000290, A000578, A271518, A272620, A275297, A275298, A275299.
%K A275300 nonn
%O A275300 0,2
%A A275300 _Zhi-Wei Sun_, Jul 22 2016