A300792 Number of ways to write n as x^2 + y^2 + z^2 + w^2, where w is a positive integer and x,y,z are nonnegative integers such that x or y or z is a square and 9*x^2 + 16*y^2 + 24*z^2 is also a square.

1, 2, 1, 2, 4, 1, 2, 2, 2, 6, 2, 3, 5, 1, 4, 1, 5, 7, 4, 5, 1, 5, 2, 1, 9, 6, 5, 3, 4, 7, 2, 2, 6, 7, 3, 5, 7, 4, 4, 6, 6, 4, 5, 3, 9, 4, 2, 1, 4, 11, 5, 9, 5, 6, 4, 1, 9, 7, 3, 6, 5, 4, 4, 2, 14, 4, 6, 5, 2, 8, 2, 7, 9, 5, 5, 4, 3, 8, 1, 4

Offset

1,2

Comments

Conjecture 1: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 16^k*m with k = 0,1,2,... and m = 1, 3, 6, 14, 21, 24, 56, 79, 119, 143, 248, 301, 383, 591, 728, 959, 1223, 1751, 2311, 6119.

Conjecture 2: Any positive integer n can be written as x^2 + y^2 + z^2 + w^2, where w is a positive integer and x,y,z are nonnegative integers such that x or y or z is a square and 36*x^2 + 40*y^2 + 45*z^2 is also a square.

See also A300791 for a similar conjecture.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..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], 2017-2018.

Example

a(6) = 1 since 6 = 1^2 + 1^2 + 0^2 + 2^2 with 1 = 1^2 and 9*1^2 + 16*1^2 + 24*0^2 = 5^2.
a(14) = 1 since 14 = 1^2 + 0^2 + 3^2 + 2^2 with 1 = 1^2 and 9*1^2 + 16*0^2 + 24*3^2 = 15^2.
a(728) = 1 since 728 = 10^2 + 0^2 + 12^2 + 22^2 with 0 = 0^2 and 9*10^2 + 16*0^2 + 24*12^2 = 66^2.
a(959) = 1 since 959 = 25^2 + 18^2 + 3^2 + 1^2 with 25 = 5^2 and 9*25^2 + 16*18^2 + 24*3^2 = 105^2.
a(1751) = 1 since 1751 = 19^2 + 25^2 + 18^2 + 21^2 with 25 = 5^2 and 9*19^2 + 16*25^2 + 24*18^2 = 145^2.
a(2311) = 1 since 2311 = 1^2 + 41^2 + 23^2 + 10^2 with 1 = 1^2 and 9*1^2 + 16*41^2 + 24*23^2 = 199^2.
a(6119) = 1 since 6119 = 1^2 + 5^2 + 3^2 + 78^2 with 1 = 1^2 and 9*1^2 + 16*5^2 + 24*3^2 = 25^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[(SQ[x]||SQ[y]||SQ[z])&&SQ[9x^2+16y^2+24z^2]&&SQ[n-x^2-y^2-z^2], r=r+1], {x, 0, Sqrt[n-1]}, {y, 0, Sqrt[n-1-x^2]}, {z, 0, Sqrt[n-1-x^2-y^2]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

Crossrefs

Cf. A000118, A000290, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300791.

Sequence in context: A193829 A337714 A198069 * A132082 A129644 A081517

Adjacent sequences: A300789 A300790 A300791 * A300793 A300794 A300795

Keyword

nonn

Author

Zhi-Wei Sun, Mar 12 2018

Status

approved