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 A275738 Number of ordered ways to write n as w^2 + x^2*(1+y^2+z^2), where w,x,y,z are nonnegative integers with x > 0, y <= z and y == z (mod 2). 4
 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 1, 3, 4, 1, 1, 2, 3, 3, 2, 5, 5, 1, 1, 1, 5, 3, 3, 5, 3, 2, 2, 1, 2, 4, 2, 7, 7, 1, 2, 3, 5, 3, 2, 3, 8, 3, 1, 3, 4, 4, 3, 9, 6, 3, 3, 1, 4, 4, 1, 6, 5, 2, 3, 2, 5, 3, 3, 5, 8, 3, 1, 3, 7, 4, 4, 8, 4, 2, 2, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Conjecture: For any n > 0, we have a(n) > 0, i.e., n can be written as w^2 + x^2*(1+(z-y)^2+(y+z)^2) = w^2 + x^2*(1+2*y^2+2*z^2), where w,x,y,z are nonnegative integers with x > 0 and y <= z. Moreover, any positive integer n not equal to 449 can be written as 4^k*(1+x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x == y (mod 2). This is stronger than Lagrange's four-square theorem, and we have verified it for n up to 10^6. See also A275648, A275656, A275675, A275676 and A275678 for similar conjectures. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.NT], 2016-2017. Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190, 2017. EXAMPLE a(2) = 1 since 2 = 1^2 + 1^2*(1+0^2+0^2) with 0 + 0 even. a(7) = 1 since 7 = 2^2 + 1^2*(1+1^2+1^2) with 1 + 1 even. a(59) = 1 since 59 = 0^2 + 1^2*(1+3^2+7^2) with 3 + 7 even. a(71) = 1 since 71 = 6^2 + 1^2*(1+3^2+5^2) with 3 + 5 even. a(113) = 2 since 113 = 7^2 + 8^2*(1+0^2+0^2) = 8^2 + 7^2*(1+0^2+0^2) with 0 + 0 even. a(143) = 1 since 143 = 6^2 + 1^2*(1+5^2+9^2) with 5 + 9 even. a(191) = 1 since 191 = 10^2 + 1^2*(1+3^2+9^2) with 3 + 9 even. a(449) = 3 since 449 = 18^2 + 5^2*(1+0^2+2^2) with 0 + 2 even, and 449 = 7^2 + 20^2*(1+0^2+0^2) = 20^2 + 7^2*(1+0^2+0^2) with 0 + 0 even. a(497) = 1 since 497 = 15^2 + 4^2*(1+0^2+4^2) with 0 + 4 even. a(2033) = 1 since 2033 = 33^2 + 4^2*(1+3^2+7^2) with 3 + 7 even. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]] Do[r=0; Do[If[SQ[n-x^2*(1+2y^2+2z^2)], r=r+1], {x, 1, Sqrt[n]}, {y, 0, Sqrt[(n/x^2-1)/4]}, {z, y, Sqrt[(n/x^2-1-2y^2)/2]}]; Print[n, " ", r]; Continue, {n, 1, 80}] CROSSREFS Cf. A000118, A000290, A271518, A275648, A275656, A275675, A275676, A275678. Sequence in context: A179617 A140188 A180050 * A202603 A279612 A261019 Adjacent sequences: A275735 A275736 A275737 * A275739 A275740 A275741 KEYWORD nonn AUTHOR Zhi-Wei Sun, Aug 07 2016 STATUS approved

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