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A271721
Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= z >= 0, x > 0 and w >= z such that (x-y)*(w-z) is a square.
37
1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 2, 4, 5, 3, 3, 3, 2, 1, 2, 3, 5, 4, 5, 2, 2, 4, 2, 3, 5, 1, 4, 4, 5, 3, 3, 4, 5, 4, 3, 4, 2, 2, 3, 3, 5, 3, 8, 4, 6, 3, 2, 4, 6, 3, 3, 4, 4, 5, 2, 3, 7, 6, 7, 2, 3, 2, 5, 6, 8, 3, 7, 3, 2, 2, 3, 6, 11, 5, 8, 5, 8, 4, 2, 3, 8, 4, 5, 5, 3, 1, 2, 9, 10, 5, 8
OFFSET
1,2
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 5, 11, 15, 23, 35, 95, 4^k*190 (k = 0,1,2,...).
(ii) For each k = 4, 5, 6, 7, 8, 11, 12, 13, 15, 17, 18, 20, 22, 25, 27, 29, 33, 37, 38, 41, 50, 61, any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that (x-y)*(w-k*z) is a square.
(iii) For each triple (a,b,c) = (3,1,1), (1,2,1), (2,2,1), (3,2,1), (2,2,2), (6,2,1), (1,3,1), (3,3,1), (15,3,1), (1,4,1), (2,4,1), (1,5,1), (3,5,1), (5,5,1), (1,5,2), (1,6,1), (2,6,1), (3,6,1), (15,6,1), (1,7,1), (5,7,1), (1,8,1), (1,8,5), (3,9,1), (1,10,1), (1,12,1), (1,13,1), (3,13,1), (1,14,1), (1,15,1), (1,15,2), (6,16,1), (2,18,1), (3,18,1), (1,20,2), (1,21,1), (3,21,1), (1,23,1), (1,24,1), (1,27,1), (3,27,1), (1,34,1), (1,45,1), (3,45,1), (3,48,1), (1,55,1), (1,60,1), (5,60,1), any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that a*(x+b*y)*(w-c*z) is a square.
This is stronger than Lagrange's four-square theorem. Note that for k = 2 or 3, any natural number n can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers and (x-y)*(w-k*z) = 0, for, if n cannot be represented by x^2 + y^2 + 2*z^2 then it has the form 4^k*(16*m+14) (k,m = 0,1,2,...) and hence it can be represented by x^2 + y^2 + (k^2+1)*z^2. It is known that natural numbers not represented by x^2 + y^2 + 5*z^2 have the form 4^k*(8*m+3), and that positive even numbers not represented by x^2 + y^2 + 10*z^2 have the form 4^k*(16*m+6) (as conjectured by S. Ramanujan and proved by L. E. Dickson).
See also A271510, A271513, A271518, A271644, A271714 and A271724 for other conjectures refining Lagrange's theorem.
REFERENCES
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33(1927), 63-70.
L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939, pp. 112-113.
LINKS
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723, 2016.
EXAMPLE
a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 0^2 with 1 = 1 > 0 = 0 and (1-1)*(0-0) = 0^2.
a(5) = 1 since 5 = 2^2 + 1^2 + 0^2 + 0^2 with 2 > 1 > 0 = 0 and (2-1)*(0-0) = 0^2.
a(11) = 1 since 11 = 1^2 + 1^2 + 0^2 + 3^2 with 1 = 1 > 0 < 3 and (1-1)*(3-0) = 0^2.
a(14) = 2 since 14 = 3^2 + 1^2 + 0^2 + 2^2 with 3 > 1 > 0 < 2 and (3-1)*(2-0) = 2^2, and also 14 = 3^2 + 2^2 + 0^2 + 1^2 with 3 > 2 > 0 < 1 and (3-2)*(1-0) = 1^2.
a(15) = 1 since 15 = 3^2 + 2^2 + 1^2 + 1^2 with 3 > 2 > 1 = 1 and (3-2)*(1-1) = 0^2.
a(23) = 1 since 23 = 3^2 + 3^2 + 1^2 + 2^2 with 3 = 3 > 1 < 2 and (3-3)*(2-1) = 0^2.
a(35) = 1 since 35 = 3^2 + 3^2 + 1^2 + 4^2 with 3 = 3 > 1 < 4 and (3-3)*(4-1) = 0^2.
a(95) = 1 since 95 = 5^2 + 5^2 + 3^2 + 6^2 with 5 = 5 > 3 < 6 and (5-5)*(6-3) = 0^2.
a(190) = 1 since 190 = 13^2 + 4^2 + 1^2 + 2^2 with 13 > 4 > 1 < 2 and (13-4)*(2-1) = 3^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(Sqrt[n-x^2-y^2-z^2]-z)*(x-y)], r=r+1], {z, 0, Sqrt[n/4]}, {y, z, Sqrt[(n-z^2)/2]}, {x, Max[1, y], Sqrt[(n-y^2-2z^2)]}]; Print[n, " ", r]; Continue, {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 12 2016
STATUS
approved