

A283617


Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers and 0 <= z <= w such that x^3 + 2*y^3 is a square.


2



1, 2, 3, 2, 3, 3, 3, 1, 2, 3, 4, 3, 3, 2, 3, 2, 2, 4, 6, 4, 6, 3, 3, 2, 2, 3, 6, 5, 3, 4, 3, 2, 3, 4, 5, 4, 7, 3, 4, 2, 3, 6, 5, 3, 3, 4, 2, 3, 2, 3, 7, 4, 6, 4, 5, 3, 2, 4, 4, 4, 4, 3, 4, 4, 3, 6, 8, 4, 9, 6, 2, 3, 4, 4, 7, 5, 4, 4, 3, 1, 3
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OFFSET

0,2


COMMENTS

Conjecture: (i) Let a and b >= a be positive integers with gcd(a,b) squarefree. Then, every n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that a*x^3 + b*y^3 is a square, if and only if (a,b) is among the ordered pairs (1,2), (1,8), (2,16), (4,23), (4,31), (5,9), (8,9), (8,225), (9,47), (25,88), (50,54).
(ii) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x a nonnegative integer and y,z,w integers such that a*x^3 + b*y^3 is a square, whenever (a,b) is among the ordered pairs (1,8), (2,16), (8,1), (9,8), (88,25), (225,8).
(iii) Any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z integers and w a positive integer such that x^3 + 19*y^3 + 19*z^3 is an integer cube.
(iv) Every n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that a*x^3 + b*y^3 + c*z^3 + d*w^3 is a square, whenever (a,b,c,d) is among the ordered quadruples (1,1,3,4), (1,2,3,5), (1,2,3,7), (1,2,4,5),(1,2,5,7), (1,3,4,5), (1,3,4,7), (1,3,6,8), (1,3,8,13), ((1,4,5,9), (1,7,8,11), (1,8,9,10), (1,8,9,11), (2,3,4,7), (2,4,7,9), (2,4,7,14), (2,7,9,11), (3,4,5,7), (3,8,9,11).
By the linked JNT paper, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + 2*y is a square.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..10000
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.
ZhiWei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.


EXAMPLE

a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 0^2 with 0^3 + 2*0^3 = 0^2.
a(7) = 1 since 7 = (1)^2 + 1^2 + 1^2 + 2^2 with (1)^3 + 2*1^3 = 1^2.
a(79) = 1 since 79 = 3^2 + 3^2 + 5^2 + 6^2 with 3^3 + 2*3^3 = 9^2.
a(88) = 1 since 88 = 4^2 + 0^2 + 6^2 + 6^2 with 4^3 + 2*0^3 = 8^2.
a(151) = 1 since 151 = (1)^2 + 1^2 + 7^2 + 10^2 with (1)^3 + 2*1^3 = 1^2.
a(219) = 1 since 219 = 1^2 + 0^2 + 7^2 + 13^2 with 1^3 + 2*0^3 = 1^2.
a(438) = 1 since 438 = (1)^2 + 1^2 + 6^2 + 20^2 with (1)^3 + 2*1^3 = 1^2.
a(471) = 1 since 471 = 3^2 + (1)^2 + 10^2 + 19^2 with 3^3 + 2*(1)^3 = 5^2.
a(599) = 1 since 599 = 7^2 + (3)^2 + 10^2 + 21^2 with 7^3 + 2*(3)^3 = 17^2.
a(751) = 1 since 751 = 3^2 + 3^2 + 2^2 + 27^2 with 3^3 + 2*3^3 = 9^2.
a(807) = 1 since 807 = 3^2 + (1)^2 + 11^2 + 26^2 with 3^3 + 2*(1)^3 = 5^2.
a(19743) = 1 since 19743 = (25)^2 + 25^2 + 58^2 + 123^2 with (25)^3 + 2*25^3 = 125^2.


MATHEMATICA

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


CROSSREFS

Cf. A000118, A000290, A000578, A271518, A281976, A283170, A283196.
Sequence in context: A244893 A321478 A076982 * A164886 A091935 A086063
Adjacent sequences: A283614 A283615 A283616 * A283618 A283619 A283620


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 12 2017


STATUS

approved



