A283269 Number of ways to write n as x^2 + y^2 + z^2 with x,y,z integers such that x + 3*y + 5*z is a square

1, 1, 2, 2, 0, 3, 5, 0, 3, 4, 4, 1, 0, 3, 7, 0, 1, 5, 3, 3, 0, 5, 3, 0, 6, 2, 8, 2, 0, 8, 3, 0, 2, 3, 6, 7, 0, 2, 6, 0, 6, 8, 4, 1, 0, 4, 3, 0, 2, 3, 7, 5, 0, 4, 13, 0, 8, 5, 2, 3, 0, 6, 6, 0, 0, 7, 13, 2, 0, 7, 3, 0, 5, 4, 9, 1, 0, 5, 3, 0, 3

Offset

0,3

Comments

Conjecture: (i) a(2^k*m) > 0 for any positive odd integers k and m. Also, a(4*n+1) = 0 only for n = 63.

(ii) For any positive odd integers k and m, we can write 2^k*m as x^2 + y^2 + z^2 with x,y,z integers such that x + 3*y + 5*z is twice a square.

(iii) For any positive odd integer n not congruent to 7 modulo 8, we can write n as x^2 + y^2 + z^2 with x,y,z integers such that x + 3*y + 4*z is a square.

(iv) Let n be any nonnegative integer. Then we can write 8*n + 1 as x^2 + y^2 + z^2 with x + 3*y a square, where x and y are integers, and z is a positive integer. Also, except for n = 2255, 4100 we can write 4*n + 2 as x^2 + y^2 + z^2 with x + 3*y a square, where x,y,z are integers.

(v) For each n = 0,1,2,..., we can write 8*n + 6 as x^2 + y^2 + z^2 with x + 2*y a square, where x,y,z are integers with y nonnegative and z odd.

The Gauss-Legendre theorem asserts that a nonnegative integer can be written as the sum of three squares if and only if it is not of the form 4^k*(8m+7) with k and m nonnegative integers. Thus a(4^k*(8m+7)) = 0 for all k,m = 0,1,2,....

See also A283273 for a similar conjecture.

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(11) = 1 since 11 = 3^2 + 1^2 + (-1)^2 with 3 + 3*1 + 5*(-1) = 1^2.
a(43) = 1 since 43 = (-5)^2 + (-3)^2 + 3^2 with (-5) + 3*(-3) + 5*3 = 1^2.
a(75) = 1 since 75 = (-1)^2 + 5^2 + 7^2 with (-1) + 3*5 + 5*7 = 7^2.
a(262) = 1 since 262 = 1^2 + 15^2 + (-6)^2 with 1 + 3*15 + 5*(-6) = 4^2.
a(277) = 1 since 277 = (-6)^2 + 4^2 + 15^2 with (-6) + 3*4 + 5*15 = 9^2.
a(617) = 1 since 617 = 17^2 + 18^2 + 2^2 with 17 + 3*18 + 5*2 = 9^2.
a(1430) = 1 since 1430 = (-13)^2 + (-6)^2 + 35^2 with (-13) + 3*(-6) + 5*35 = 12^2.
a(5272) = 1 since 5272 = (-66)^2 + 30^2 + (-4)^2 with (-66) + 3*30 + 5*(-4) = 2^2.
a(7630) = 1 since 7630 = (-78)^2 + 39^2 + 5^2 with (-78) + 3*39 + 5*5 = 8^2.
a(7933) = 1 since 7933 = (-56)^2 + 69^2 + (-6)^2 with (-56) + 3*69 + 5*(-6) = 11^2.
a(14193) = 1 since 14193 = (-7)^2 + 112^2 + 40^2 with (-7) + 3*112 + 5*40 = 23^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[n-x^2-y^2]&&SQ[(-1)^i*x+(-1)^j*3y+(-1)^k*5*Sqrt[n-x^2-y^2]], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {i, 0, Min[x, 1]}, {j, 0, Min[y, 1]}, {k, 0, Min[Sqrt[n-x^2-y^2], 1]}]; Print[n, " ", r]; Continue, {n, 0, 80}]

Crossrefs

Cf. A000290, A005875, A271518, A283170, A283196, A283204, A283205, A283239, A283273.

Sequence in context: A077264 A349344 A188333 * A201947 A098816 A214639

Adjacent sequences: A283266 A283267 A283268 * A283270 A283271 A283272

Keyword

nonn

Author

Zhi-Wei Sun, Mar 04 2017

Status

approved