

A273458


Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with xy+z+w a nonnegative cube, where x,y,z,w are integers with x >= y >= 0 and x >= z <= w.


8



1, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 1, 5, 4, 3, 2, 1, 4, 3, 3, 6, 3, 2, 5, 3, 9, 3, 1, 1, 7, 5, 3, 7, 10, 4, 6, 2, 10, 2, 6, 2, 12, 7, 2, 5, 9, 3, 3, 6, 13, 3, 8, 3, 18, 3, 8, 5, 7, 3, 3, 5, 13, 8, 5, 3, 19, 4, 7, 7, 16, 1, 11, 5, 14, 7, 2, 3, 12, 5, 4
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OFFSET

0,2


COMMENTS

Conjecture: a(n) > 0 for all n = 0,1,2,....
In the latest version of arXiv:1605.03074, the authors showed that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + y + z + w is a cube (or a square).
For more conjectural refinements of Lagrange's foursquare theorem, see the author's preprint arXiv:1604.06723.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..10000
YuChen Sun and ZhiWei Sun, Two refinements of Lagrange's foursquare theorem, arXiv:1605.03074 [math.NT], 2016.
ZhiWei Sun, Refining Lagrange's foursquare theorem, arXiv:1604.06723 [math.GM], 2016.


EXAMPLE

a(12) = 1 since 12 = 3^2 + 1^2 + (1)^2 + (1)^2 with 3  1 + (1) + (1) = 0^3.
a(17) = 1 since 17 = 2^2 + 0^2 + 2^2 + (3)^2 with 2  0 + 2 + (3) = 1^3.
a(28) = 1 since 28 = 3^2 + 1^2 + 3^2 + 3^2 with 3  1 + 3 + 3 = 2^3.
a(29) = 1 since 29 = 3^2 + 0^2 + 2^2 + (4)^2 with 3  0 + 2 + (4) = 1^3.
a(71) = 1 since 71 = 5^2 + 1^2 + 3^2 + (6)^2 with 5  1 + 3 + (6) = 1^3.
a(149) = 1 since 149 = 8^2 + 0^2 + 2^2 + (9)^2 with 8  0 + 2 + (9) = 1^3.
a(188) = 1 since 188 = 13^2 + 3^2 + 1^2 + (3)^2 with 13  3 + 1 + (3) = 2^3.
a(284) = 1 since 284 = 15^2 + 5^2 + 3^2 + (5)^2 with 15  5 + 3 + (5) = 2^3.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
CQ[n_]:=CQ[n]=n>=0&&IntegerQ[n^(1/3)]
Do[r=0; Do[If[SQ[nx^2y^2z^2]&&CQ[xy+(1)^j*z+(1)^k*Sqrt[nx^2y^2z^2]], r=r+1], {y, 0, (n/2)^(1/2)}, {x, y, Sqrt[ny^2]}, {z, 0, Min[x, Sqrt[(nx^2y^2)/2]]}, {j, 0, Min[1, z]}, {k, 0, Min[1, Sqrt[nx^2y^2z^2]]}];
Print[n, " ", r]; Continue, {n, 0, 80}]


CROSSREFS

Cf. A000118, A000290, A000578, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273429, A273432, A273568.
Sequence in context: A268616 A002233 A241516 * A159953 A074595 A084126
Adjacent sequences: A273455 A273456 A273457 * A273459 A273460 A273461


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 22 2016


STATUS

approved



