OFFSET
1,2
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m (k = 0,1,2,... and m = 1, 3, 11, 43, 547, 763, 1739, 6783).
(ii) For each quadruples (a,b,c,d) = (1,3,78,27), (1,3,222,75), (4,12,81,108), (6,27,25,75), (7,21,112,32), any positive integer can be written as w^2 + x^2 + y^2 + z^2 with a*w^2*x^2 + b*x^2*y^2 + c*y^2*z^2 + d*z^2*w^2 a square, where w is a positive integer and x,y,z are integers.
(iii) Each n = 0,1,2,.... can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that w^2*x^2 + 4*x^2*y^2 + 44*y^2*z^2 + 16*z^2*w^2 = 5*t^2 for some integer t.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arxiv:1604.06723, 2016.
EXAMPLE
a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 > 0 and 1^2*0^2 + 5*0^2*0^2 + 80*0^2*0^2 + 20*0^2*1^2 = 0^2.
a(2) = 2 since 2 = 1^2 + 0^2 + 1^2 + 0^2 with 1 > 0 and 1^2*0^2 + 5*0^2*1^2 + 80*1^2*0^2 + 20*0^2*1^2 = 0^2, and also 2 = 1^2 + 1^2 + 0^2 + 0^2 with 1 > 0 and 1^2*1^2 + 5*1^2*0^2 + 80*0^2*0^2 + 20*0^2*1^2 = 1^2.
a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 with 1 > 0 and 1^2*0^2 + 5*0^2*1^2 + 80*1^2*1^2 + 20*1^2*1^2 = 10^2.
a(11) = 1 since 11 = 1^2 + 0^2 + 1^2 + 3^2 with 1 > 0 and
1^2*0^2 + 5*0^2*1^2 + 80*1^2*3^2 + 20*3^2*1^2 = 30^2.
a(43) = 1 since 43 = 3^2 + 0^2 + 3^2 + 5^2 with 3 > 0 and 3*0^2 + 5*0^2*3^2 + 80*3^2*5^2 + 20*5^2*3^2 = 150^2.
a(547) = 1 since 547 = 3^2 + 0^2 + 3^2 + 23^2 with 3 > 0 and 3^2*0^2 + 5*0^2*3^2 + 80*3^2*23^2 + 20*23^2*3^2 = 690^2.
a(763) = 1 since 763 = 13^2 + 20^2 + 13^2 + 5^2 with 13 > 0 and 13^2*20^2 + 5*20^2*13^2 + 80*13^2*5^2 + 20*5^2*13^2 = 910^2.
a(1739) = 1 since 1739 = 15^2 + 16^2 + 27^2 + 23^2 with 15 > 0 and 15^2*16^2 + 5*16^2*27^2 + 80*27^2*23^2 + 20*23^2*15^2 = 5850^2.
a(6783) = 1 since 6783 = 17^2 + 73^2 + 18^2 + 29^2 with 17 > 0 and 17^2*73^2 + 5*73^2*18^2 + 80*18^2*29^2 + 20*29^2*17^2 = 6069^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(n-x^2-y^2-z^2)*x^2+5*x^2*y^2+80*y^2*z^2+20*z^2*(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]}]; Print[n, " ", r]; Continue, {n, 1, 70}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 17 2016
STATUS
approved