

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.


2



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, 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, 4, 11, 9, 8, 6, 10, 13, 1, 7, 10, 8, 8, 2, 10, 14, 3, 10
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OFFSET

0,2


COMMENTS

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.
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.
See also A275297, A275298, A275299 and A272620 for similar conjectures.


LINKS

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


EXAMPLE

a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 0^3 with 0 + 0 + 0 = 0^2 and 0 = 0 = 0.
a(5) = 1 since 5 = 2^2 + 0^2 + (1)^2 + 0^3 with 2 + 0 + (1) = 1^2 and 2 > 0 < 1.
a(12) = 1 since 12 = 3^2 + (1)^2 + (1)^2 + 1^3 with 3 + (1) + (1) = 1^2 and 3 > 1 = 1.
a(20) = 1 since 20 = 3^2 + 1^2 + (3)^2 + 1^3 with 3 + 1 + (3) = 1^2 and 3 > 1 < 3.
a(23) = 1 since 23 = 3^2 + (2)^2 + 3^2 + 1^3 with 3 + (2) + 3 = 2^2 and 3 > 2 < 3.
a(31) = 1 since 31 = 5^2 + 1^2 + (2)^2 + 1^3 with 5 + 1 + (2) = 2^2 and 5 > 1 < 2.
a(47) = 1 since 47 = 6^2 + 1^2 + (3)^2 + 1^3 with 6 + 1 + (3) = 2^2 and 6 > 1 < 3.
a(71) = 1 since 71 = 6^2 + 3^2 + (5)^2 + 1^3 with 6 + 3 + (5) = 2^2 and 6 > 3 < 5.
a(103) = 1 since 103 = 7^2 + 2^2 + 7^2 + 1^3 with 7 + 2 + 7 = 4^2 and 7 > 2 < 7.
a(148) = 1 since 148 = 9^2 + (2)^2 + (6)^2 + 3^3 with 9 + (2) + (6) = 1^2 and 9 > 2 < 6.
a(164) = 1 since 164 = 9^2 + 1^2 + (9)^2 + 1^3 with 9 + 1 + (9) = 1^2 and 9 > 1 < 9.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[nw^3y^2z^2]&&SQ[Sqrt[nw^3y^2z^2]+(1)^i*y+(1)^j*z], r=r+1], {w, 0, n^(1/3)}, {y, 0, Sqrt[(nw^3)/3]}, {i, 0, Min[1, y]}, {z, y, Sqrt[nw^32y^2]}, {j, 0, Min[1, z]}]; Print[n, " ", r]; Continue, {n, 0, 80}]


CROSSREFS

Cf. A000290, A000578, A271518, A272620, A275297, A275298, A275299.
Sequence in context: A215409 A239232 A153012 * A283833 A280759 A016651
Adjacent sequences: A275297 A275298 A275299 * A275301 A275302 A275303


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jul 22 2016


STATUS

approved



