This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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 Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. Zhi-Wei 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[n-x^2-y^2-z^2], r=r+1], {z, 0, Sqrt[(n-x^2-y^2)/2]}]], {x, 0, Sqrt[n]}, {i, 0, Min[x, 1]}, {y, 0, Sqrt[n-x^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 Zhi-Wei Sun, Mar 12 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 25 22:10 EDT 2019. Contains 324357 sequences. (Running on oeis4.)