A271778 -id:A271778 - cp's OEIS frontend

A271824 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+2y)^2 + 8z^2 + 40*w^2 a square, where x is a positive integer and y,z,w are nonnegative integers.

1, 2, 2, 1, 2, 2, 2, 2, 1, 4, 1, 3, 3, 2, 1, 1, 3, 6, 3, 3, 4, 1, 1, 2, 3, 4, 3, 3, 2, 5, 4, 2, 1, 3, 3, 3, 5, 1, 5, 4, 2, 6, 3, 2, 5, 3, 3, 3, 2, 8, 3, 6, 6, 4, 4, 2, 4, 6, 3, 3, 5, 3, 4, 1, 5, 5, 4, 4, 2, 6, 1, 6, 2, 4, 7, 4, 3, 5, 7, 3

Offset: 1

Zhi-Wei Sun, Apr 14 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 9, 11, 15, 23, 33, 71, 129, 167, 187, 473, 4^k*m (k = 0,1,2,... and m = 1, 22, 38, 278). Also, any positive integer can be written as x^2 + y^2 + z^2 + w^2 with 9*(x+2*y)^2 + 16*z^2 + 24*w^2 a square, where x is a positive integer and y,z,w are nonnegative integers.
(ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and (a*x-b*y)^2 + c*z^2 + d*w^2 a square, provided that (a,b,c,d) is among the quadruples (4,8,1,8), (12,24,1,24), (2,4,5,40), (3,6,7,9), (3,9,7,9), (3,6,7,63), (1,2,8,16), (1,2,8,40), (3,6,8,40), (2,6,9,12), (3,5,9,15), (4,8,9,16), (12,24,9,16), (3,6,15,25), (3,6,16,24), (3,12,16,24), (6,9,16,24), (9,12,16,24), (4,8,16,41), (8,12,16,41), (3,6,16,48), (6,9,16,48), (2,3,16,56), (3,6,28,63), (2,4,36,45), (6,12,40,45), (7,14,56,64) and (2,6,57,60).
(iii) Let a and be positive integers with a <= b and gcd(a,b) squarefree. Then any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and (a*x+b*y)*z a square, if and only if (a,b) is among the ordered pairs (1,1), (1,2), (1,3), (2,5), (3,3), (3,6), (3,15), (5,6), (5,11), (5,13), (6,15), (8,46) and (9,23).
(iv) Let a and b be positive integers with a <= b and gcd(a,b) squarefree. Then, any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and (a*x^2+b*y^2)*z a square, if and only if (a,b) is among the ordered pairs (3,13), (5,11), (15,57), (15,165) and (138,150).
There are many ordered pairs (a,b) of integers with gcd(a,b) squarefree such that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers and a*x^2 + b*y^2 a square. For example, we have shown that (1,-1), (2,-2), (3,-3) and (1,2) are indeed such ordered pairs.
See also A271510, A271513, A271518, A271665, A271714, A271721, A271724, A271775 and A271778 for other conjectures refining Lagrange's four-square theorem.

			a(9) = 1 since 9 = 3^2 + 0^2 + 0^2 + 0^2 with (3+2*0)^2 + 8*0^2 + 40*02 = 3^2.
a(11) = 1 since 11 = 1^2 + 1^2 + 3^2 + 0^2 with (1+2*1)^2 + 8*3^2 + 40*0^2 = 9^2.
a(15) = 1 since 15 = 1^2 + 3^2 + 2^2 + 1^2 with (1+2*3)^2 + 8*2^2 + 40*1^2 = 11^2.
a(22) = 1 since 22 = 3^2 + 2^2 + 3^2 + 0^2 with (3+2*2)^2 + 8*3^2 + 40*0^2 = 11^2.
a(23) = 1 since 23 = 1^2 + 3^2 + 2^2 + 3^2 with (1+2*3)^2 + 8*2^2 + 40*3^2 = 21^2.
a(33) = 1 since 33 = 4^2 + 1^2 + 0^2 + 4^2 with (4+2*1)^2 + 8*0^2 + 40*4^2 = 26^2.
a(38) = 1 since 38 = 5^2 + 2^2 + 0^2 + 3^2 with (5+2*2)^2 + 8*0^2 + 40*3^2 = 21^2.
a(71) = 1 since 71 = 1^2 + 6^2 + 5^2 + 3^2 since (1+2*6)^2 + 8*5^2 + 40*3^2 = 27^2.
a(129) = 1 since 129 = 5^2 + 6^2 + 8^2 + 2^2 with (5+2*6)^2 + 8*8^2 + 40*2^2 = 31^2.
a(167) = 1 since 167 = 11^2 + 1^2 + 3^2 + 6^2 with (11+2*1)^2 + 8*3^2 + 40*6^2 = 41^2.
a(187) = 1 since 187 = 3^2 + 5^2 + 12^2 + 3^2 with (3+2*5)^2 + 8*12^2 + 40*3^2 = 41^2.
a(278) = 1 since 278 = 3^2 + 0^2 + 10^2 + 13^2 with (3+2*0)^2 + 8*10^2 + 40*13^2 = 87^2.
a(473) = 1 since 473 = 7^2 + 10^2 + 0^2 + 18^2 with (7+2*10)^2 + 8*0^2 + 40*18^2 = 117^2.

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.

Cf. A000118, A000290, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(x+2y)^2+8z^2+40(n-x^2-y^2-z^2)],r=r+1],{x,1,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A262357 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w^2x^2 + 5x^2y^2 + 80y^2z^2 + 20z^2*w^2 a square, where w is a positive integer and x,y,z are nonnegative integers.

Original entry on oeis.org

1, 2, 1, 1, 4, 3, 2, 2, 4, 5, 1, 1, 6, 3, 2, 1, 6, 7, 2, 4, 8, 6, 2, 3, 8, 9, 3, 2, 8, 5, 2, 2, 6, 6, 2, 4, 9, 5, 4, 5, 8, 5, 1, 1, 10, 5, 3, 1, 5, 9, 3, 6, 10, 10, 6, 3, 5, 5, 2, 2, 12, 3, 5, 1, 13, 9, 3, 6, 10, 9

Offset: 1

Zhi-Wei Sun, Apr 17 2016

nonn

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.
See also A268507, A269400, A271510, A271513, A271518, A271665, A271714, A271721, A271724, A271775, A271778 and A271824 for other conjectures refining Lagrange's four-square theorem.

			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.

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.

Cf. A000118, A000290, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824.

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}]

A268507 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w > 0, w >= x <= y <= z such that x^2y^2 + y^2z^2 + z^2*x^2 is a square, where w,x,y,z are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 16 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 2^k, 4^k*m (k = 0,1,2,... and m = 3, 7, 23, 31, 39, 47, 55, 71, 79, 151, 191, 551).
(ii) For each triple (a,b,c) = (1,4,4), (1,4,16), (1,4,26), (1,4,31), (1,4,34), (1,9,9), (1,9,11), (1,9,17), (1,9,21), (1,9,27), (1,9,33), (1,9,41), (1,18,24), (1,36,44), (3,4,8), (4,6,9), (4,8,19), (4,8,27), (4,9,36), (4,16,41), (4,19,29), (5,9,25), (7,9,33), (7,25,49), (9,10,45), (9,12,28), (9,16,36), (9,21,49), (9,24,37), (9,25,27), (9,25,45), (9,30,40), (9,32,64), (9,34,36), (9,44,61), (14,25,40), (16,17,36), (16,20,25), (24,36,39), (25,40,64), (25,45,51), (27,36,37), (28,44,49), (32,49,64), (36,43,45), (36,54,58), any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that a*x^2*y^2 + b*y^2*z^2 + c*z^2*x^2 is a square.
See also A269400, A271510, A271513, A271518, A271665, A271714, A271721, A271724, A271775, A271778 and A271824 for other conjectures refining Lagrange's four-square theorem.
The author has proved in arXiv:1604.06723 that a(n) > 0 for any positive integer n. - Zhi-Wei Sun, May 09 2016

			a(2) = 1 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 > 0 = 0 < 1 and 0^2*0^2 + 0^2*1^2 + 1^2*0^2 = 0^2.
a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 with 1 > 0 < 1 = 1 and 0^2*1^2 + 1^2*1^2 + 1^2*0^2 = 1^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 = 1 = 1 < 2 and 1^2*1^2 + 1^2*2^2 + 2^2*1^2 = 3^2.
a(23) = 1 since 23 = 3^2 + 1^2 + 2^2 + 3^2 with 3 > 1 < 2 < 3 and 1^2*2^2 + 2^2*3^2 + 3^2*1^2 = 7^2.
a(31) = 1 since 31 = 5^2 + 1^2 + 1^2 + 2^2 with 5 > 1 = 1 < 2 and 1^2*1^2 + 1^2*2^2 + 2^2*1^2 = 3^2.
a(39) = 1 since 39 = 5^2 + 1^2 + 2^2 + 3^2 with 5 > 1 < 2 < 3 and 1^2*2^2 + 2^2*3^2 + 3^2*1^2 = 7^2.
a(47) = 1 since 47 = 3^2 + 2^2 + 3^2 + 5^2 with 3 > 2 < 3 < 5 and 2^2*3^2 + 3^2*5^2 + 5^2*2^2 = 19^2.
a(55) = 1 since 55 = 7^2 + 1^2 + 1^2 + 2^2 with 7 > 1 = 1 < 2 and 1^2*1^2 + 1^2*2^2 + 2^2*1^2 = 3^2.
a(71) = 1 since 71 = 3^2 + 1^2 + 5^2 + 6^2 with 3 > 1 < 5 < 6 and 1^2*5^2 + 5^2*6^2 + 6^2*1^2 = 31^2.
a(79) = 1 since 79 = 5^2 + 3^2 + 3^2 + 6^2 with 5 > 3 = 3 < 6 and 3^2*3^2 + 3^2*6^2 + 6^2*3^2 = 27^2.
a(151) = 1 since 151 = 5^2 + 3^2 + 6^2 + 9^2 with 5 > 3 < 6 < 9 and 3^2*6^2 + 6^2*9^2 + 9^2*3^2 = 63^2.
a(191) = 1 since 191 = 3^2 + 1^2 + 9^2 + 10^2 with 3 > 1 < 9 < 10 and 1^2*9^2 + 9^2*10^2 + 10^2*1^2 = 91^2.
a(551) = 1 since 551 = 15^2 + 3^2 + 11^2 + 14^2 with 15 > 3 < 11 < 14 and 3^2*11^2 + 11^2*14^2 + 14^2*3^2 = 163^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.NT], 2016-2017.

Cf. A000118, A000290, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
TQ[n_]:=TQ[n]=n>0&&SQ[n]
Do[r=0;Do[If[TQ[n-x^2-y^2-z^2]&&SQ[x^2*y^2+y^2*z^2+z^2*x^2],r=r+1],{x,0,Sqrt[n/4]},{y,x,Sqrt[(n-2x^2)/2]},{z,y,Sqrt[n-2x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A269400 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 6w^2x^2 + 12x^2y^2 + 52y^2z^2 + 27z^2w^2 a square, where w,x,y are nonnegative integers and z is a positive integer.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 2, 3, 4, 3, 1, 1, 4, 5, 2, 3, 3, 4, 4, 2, 5, 5, 2, 5, 5, 2, 1, 1, 3, 6, 2, 3, 4, 8, 1, 3, 8, 7, 3, 3, 4, 5, 2, 3, 6, 9, 4, 6, 10, 4, 3, 3, 3, 8, 5, 4, 5, 5, 5, 1, 7, 4, 2, 7, 4, 5, 1, 5, 7, 5, 2, 4, 8, 1, 1, 3

Offset: 1

Zhi-Wei Sun, Apr 16 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 3, 15, 31, 39, 71, 79, 195, 311, 319, 403, 559, 591, 683, 719, 1031, 1439, 1643, 2519, 6879, 2^k, 2^(2k+1)*39 (k = 0,1,2,...). Also, any positive integer can be written as w^2 + x^2 + y^2 + z^2 with x a positive integer and w,y,z nonnegative integer such that 6*w^2*x^2 + 12*x^2*y^2 + 52*y^2*z^2 + 27*z^2*w^2 is a square.
(ii) For each triple (a,b,c) = (1,3,2), (1,11,9), (1,14,4),(1,20,25), (1,27,18), (1,36,9), (1,56,4), (4,32,25), (9,15,25), (9,35,25), (25,8,64), (25,15,54), (25,32,28), (25,35,49), (28,32,49), any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that a*w^2*x^2 + b*x^2*y^2 + c*y^2*z^2 is a square.
See also A268507, A271510, A271513, A271518, A271665, A271714, A271721, A271724, A271775, A271778 and A271824 for other conjectures refining Lagrange's four-square theorem.

			a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 1 > 0 and 6*0^2*0^2 + 12*0^2*0^2 + 52*0^2*1^2 + 27*1^2*0^2 = 0^2.
a(2) = 1 since 2 = 0^2 + 1^2 + 0^2 + 1^2 with 1 > 0 and 6*0^2*1^2 + 12*1^2*0^2 + 52*0^2*1^2 + 27*1^2*0^2 = 0^2.
a(3) = 1 since 3 = 0^2 + 1^2 + 1^2 + 1^2 with 1 > 0 and 6*0^2*1^2 + 12*1^2*1^2 + 52*1^2*1^2 + 27*1^2*0^2 = 8^2.
a(15) = 1 since 15 = 2^2 + 3^2 + 1^2 + 1^2 with 1 > 0 and 6*2^2*3^2 + 12*3^2*1^2 + 52*1^2*1^2 = 22^2.
a(31) = 1 since 31 = 1^2 + 1^2 + 2^2 + 5^2 with 5 > 0 and 6*1^2*1^2 + 12*1^2*2^2 + 52*2^2*5^2 + 27*5^2*1^2 = 77^2.
a(39) = 1 since 39 = 2^2 + 1^2 + 5^2 + 3^2 with 3 > 0 and 6*2^2*1^2 + 12*1^2*5^2 + 52*5^2*3^2 + 27*3^2*2^2 = 114^2.
a(71) = 1 since 71 = 3^2 + 1^2 + 6^2 + 5^2 with 5 > 0 and 6*3^2*1^2 + 12*1^2*6^2 + 52*6^2*5^2 + 27*5^2*3^2 = 231^2.
a(78) = 1 since 78 = 2^2 + 7^2 + 4^2 + 3^2 with 3 > 0 and 6*2^2*7^2 + 12*7^2*4^2 + 52*7^2*4^2 + 27*3^2*2^2 = 138^2.
a(79) = 1 since 79 = 2^2 + 5^2 + 7^2 + 1^2 with 1 > 0 and 6*2^2*5^2 + 12*5^2*7^2 + 52*7^2*1^2 + 27*1^2*2^2 = 134^2.
a(195) = 1 since 195 = 3^2 + 7^2 + 4^2 + 11^2 with 11 > 0 and 6*3^2*7^2 + 12*7^2*4^2 + 52*4^2*11^2 + 27*11^2*3^2 = 377^2.
a(311) = 1 since 311 = 14^2 + 9^2 + 3^2 + 5^2 with 5 > 0 and 6*14^2*9^2 + 12*9^2*3^2 + 52*3^2*5^2 + 27*5^2*14^2 = 498^2.
a(319) = 1 since 319 = 6^2 + 3^2 + 7^2 + 15^2 with 15 > 0 and 6*6^2*3^2 + 12*3^2*7^2 + 52*7^2*15^2 + 27*15^2*6^2 = 894^2.
a(403) = 1 since 403 = 3^2 + 13^2 + 12^2 + 9^2 with 9 > 0 and 6*3^2*13^2 + 12*13^2*12^2 + 52*12^2*9^2 + 27*9^2*3^2 = 963^2.
a(559) = 1 since 559 = 5^2 + 23^2 + 2^2 + 1^2 with 1 > 0 and 6*5^2*23^2 + 12*23^2*2^2 + 52*2^2*1^2 + 27*1^2*5^2 = 325^2.
a(591) = 1 since 591 = 21^2 + 11^2 + 2^2 + 5^2 with 5 > 0 and 6*21^2*11^2 + 12*11^2*2^2 + 52*2^2*5^2 + 27*5^2*21^2 = 793^2.
a(683) = 1 since 683 = 0^2 + 11^2 + 21^2 + 11^2 with 11 > 0 and 6*0^2*11^2 + 12*11^2*21^2 + 52*21^2*11^2 + 27*11^2*0^2 = 1848^2.
a(719) = 1 since 719 = 10^2 + 3^2 + 21^2 + 13^2 with 13 > 0 and 6*10^2*3^2 + 12*3^2*21^2 + 52*21^2*13^2 + 27*13^2*10^2 = 2094^2.
a(1031) = 1 since 1031 = 26^2 + 15^2 + 9^2 + 7^2 with 7 > 0 and 6*26^2*15^2 + 12*15^2*9^2 + 52*9^2*7^2 + 27*7^2*26^2 = 1494^2.
a(1439) = 1 since 1439 = 13^2 + 27^2 + 10^2 + 21^2 with 21 > 0 and 6*13^2*27^2 + 12*27^2*10^2 + 52*10^2*21^2 + 27*21^2*13^2 = 2433^2.
a(1643) = 1 since 1643 = 36^2 + 17^2 + 3^2 + 7^2 with 7 > 0 and 6*36^2*17^2 + 12*17^2*3^2 + 52*3^2*7^2 + 27*7^2*36^2 = 2004^2.
a(2519) = 1 since 2519 = 27^2 + 7^2 + 30^2 + 29^2 with 29 > 0 and 6*27^2*7^2 + 12*7^2*30^2 + 52*30^2*29^2 + 27*29^2*27^2 = 7527^2.
a(6879) = 1 since 6879 = 38^2 + 53^2 + 49^2 + 15^2 with 15 > 0 and 6*38^2*53^2 + 12*53^2*49^2 + 52*49^2*15^2 + 27*15^2*38^2 = 11922^2.

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.

Cf. A000118, A000290, A268507, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[6*(n-x^2-y^2-z^2)*x^2+12*x^2*y^2+52*y^2*z^2+27*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,1,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A272084 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 4x^2 + 5y^2 + 20zw a square, where x,y,z,w are nonnegative integers with z < w.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 19 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 2^k*q (k = 0,1,2,... and q = 1, 3, 7), 4^k*m (k = 0,1,2,... and m = 21, 30, 38, 62, 70, 77, 142, 217, 237, 302, 382, 406, 453, 670).
(ii) For each triple (a,b,c) = (1,8,8), (7,9,-12), (9,40,-24), (9,40,-60), any positive integer can be written as x^2 + y^2 + z^2 + w^2 with a*x^2 + b*y^2 + c*z*w a square, where w is a positive integer and x,y,z are nonnegative integers.
(iii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with (3*x+5*y)^2 -24*z*w a square, where x,y,z,w are nonnegative integers.  Also, for each ordered pair (a,b) = (1,4), (1,8), (1,12), (1,24), (1,32), (1,48), (25,24), (1,-4), (9,-4), (121,-20), every natural number can be written as x^2 + y^2 + z^2 + w^2 with a*x^2 + b*y*z a square, where x,y,z,w are nonnegative integers.
(iv) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with (x^2-y^2)*(w^2-2*z^2) (or (x^2-y^2)*(2*w^2-z^2) or (x^2-y^2)*(w^2-5*z^2)) a square, where w,x,y,z are integers.
See also A262357, A268507, A269400, A271510, A271513, A271518, A271665, A271714, A271721, A271724, A271775, A271778 and A271824 for other conjectures refining Lagrange's four-square theorem.

			a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 < 1 and 4*0^2 + 5*0^2 + 20*0*1 = 0^2.
a(2) = 1 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 0 < 1 and 4*1^2 + 5*0^2 + 20*0*1 = 2^2.
a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 0 < 1 and 4*1^2 + 5*1^2 + 20*0*1 = 3^2.
a(6) = 1 since 6 = 1^2 + 1^2 + 0^2 + 2^2 with 0 < 2 and 4*1^2 + 5*1^2 + 20*0*2 = 3^2.
a(14) = 1 since 14 = 1^2 + 3^2 + 0^2 + 2^2 with 0 < 2 and 4*1^2 + 5*3^2 + 20*0*2 = 7^2.
a(21) = 1 since 21 = 0^2 + 2^2 + 1^2 + 4^2 with 1 < 4 and 4*0^2 + 5*2^2 + 20*1*4 = 10^2.
a(30) = 1 since 30 = 4^2 + 2^2 + 1^2 + 3^2 with 1 < 3 and 4*4^2 + 5*2^2 + 20*1*3 = 12^2.
a(38) = 1 since 38 = 1^2 + 1^2 + 0^2 + 6^2 with 0 < 6 and 4*1^2 + 5*1^2 + 20*0*6 = 3^2.
a(62) = 1 since 62 = 1^2 + 3^2 + 4^2 + 6^2 with 4 < 6 and 4*1^2 + 5*3^2 + 20*4*6 = 23^2.
a(70) = 1 since 70 = 7^2 + 1^2 + 2^2 + 4^2 with 2 < 4 and 4*7^2 + 5*1^2 + 20*2*4 = 19^2.
a(77) = 1 since 77 = 4^2 + 6^2 + 3^2 + 4^2 with 3 < 4 and 4*4^2 + 5*6^2 + 20*3*4 = 22^2.
a(142) = 1 since 142 = 4^2 + 6^2 + 3^2 + 9^2 with 3 < 9 and 4*4^2 + 5*6^2 + 20*3*9 = 28^2.
a(217) = 1 since 217 = 6^2 + 6^2 + 8^2 + 9^2 with 8 < 9 and 4*6^2 + 5*6^2 + 20*8*9 = 42^2.
a(237) = 1 since 237 = 5^2 + 8^2 + 2^2 + 12^2 with 2 < 12 and 4*5^2 + 5*8^2 + 20*2*12 = 30^2.
a(302) = 1 since 302 = 11^2 + 9^2 + 6^2 + 8^2 with 6 < 8 and 4*11^2 + 5*9^2 + 20*6*8 = 43^2.
a(382) = 1 since 382 = 11^2 + 7^2 + 4^2 + 14^2 with 4 < 14 and 4*11^2 + 5*4^2 + 20*4*14 = 43^2.
a(406) = 1 since 406 = 8^2 + 6^2 + 9^2 + 15^2 with 9 < 15 and 4*8^2 + 5*6^2 + 20*9*15 = 56^2.
a(453) = 1 since 453 = 8^2 + 14^2 + 7^2 + 12^2 with 7 < 12 and 4*8^2 + 5*14^2 + 20*7*12 = 54^2.
a(670) = 1 since 670 = 17^2 + 11^2 + 2^2 + 16^2 with 2 < 16 and 4*17^2 + 5*11^2 + 20*2*16 = 49^2.

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.

Cf. A000118, A000290, A262357, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[4x^2+5y^2+20*z*Sqrt[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)/2]}];Print[n," ",r];Continue,{n,1,100}]

A272332 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 36x^2y + 12y^2z + z^2*x a square, where w is a positive integer and x,y,z are nonnegative integers.

Original entry on oeis.org

1, 3, 2, 2, 6, 4, 3, 3, 3, 8, 5, 2, 6, 6, 4, 1, 7, 10, 6, 8, 8, 5, 2, 2, 7, 16, 8, 3, 12, 6, 4, 3, 6, 13, 8, 8, 8, 6, 5, 7, 15, 14, 4, 2, 12, 7, 3, 2, 5, 18, 8, 12, 14, 8, 7, 4, 6, 8, 7, 5, 14, 8, 5, 2, 12, 18, 8, 12, 10, 6, 3, 5, 10, 19, 10, 3, 8, 3, 1, 6

Offset: 1

Zhi-Wei Sun, Apr 26 2016

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 16^k*m (k = 0,1,2,... and m = 1, 79, 591, 599, 1752, 1839, 10264).
We have verified that a(n) > 0 for all n = 1,...,400000.
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.

			a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 > 0 and 36*0^2*0 + 12*0^2*0 + 0^2*0 = 0^2.
a(79) = 1 since 79 = 7^2 + 1^2 + 5^2 + 2^2 with 7 > 0 and 36*1^2*5 + 12*5^2*2 + 2^2*1 = 28^2.
a(591) = 1 since 591 = 23^2 + 1^2 + 6^2 + 5^2 with 23 > 0 and 36*1^2*6 + 12*6^2*5 + 5^2*1 = 49^2.
a(599) = 1 since 599 = 6^2 + 1^2 + 11^2 + 21^2 with 6 > 0 and 36*1^2*11 + 12*11^2*21 + 21^2*1 = 177^2.
a(1752) = 1 since 1752 = 10^2 + 4^2 + 40^2 + 6^2 with 10 > 0 and 36*4^2*40 + 12*40^2*6 + 6^2*10 = 372^2.
a(1839) = 1 since 1839 = 17^2 + 37^2 + 9^2 + 10^2 with 17 > 0 and 36*37^2*9 + 12*9^2*10 + 10^2*37 = 676^2.
a(10264) = 1 since 10264 = 96^2 + 30^2 + 2^2 + 12^2 with 96 > 0 and 36*30^2*2 + 12*2^2*12 + 12^2*30 = 264^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016.

Cf. A000118, A000290, A262357, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272336.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[36*x^2*y+12*y^2*z+z^2*x],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,1,80}]

A272351 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with x^4 + 8yz*(y^2+z^2) a fourth power, where w,x,y,z are nonnegative integers with w >= x and y > z.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 2, 3, 2, 1, 4, 5, 2, 2, 3, 3, 2, 2, 3, 5, 4, 1, 5, 6, 1, 1, 5, 4, 5, 3, 2, 5, 2, 3, 7, 7, 3, 2, 5, 4, 2, 1, 5, 8, 7, 2, 5, 9, 1, 3, 4, 4, 5, 2, 5, 8, 6, 1, 8, 8, 4, 4, 6, 5, 1, 5, 5, 10, 6, 2, 6, 8, 1, 2

Offset: 1

Zhi-Wei Sun, Apr 27 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 2^k, 4^k*m (k = 0,1,2,... and m = 3, 7, 31, 55, 71, 79, 151, 191).
(ii) For (b,c) = (8,8), (16,64), any positive integer can be written as w^2 + x^2 + y^2 + z^2 with x^4 + b*y^3*z + c*y*z^3 a fourth power, where w is a positive integer and x,y,z are nonnegative integers.
(iii) For each triple (a,b,c) = (1,20,60), (1,24,56), (9,20,60), (9,32,96), any positive integer can be written as w^2 + x^2 + y^2 + z^2 with a*x^4 + b*y^3*z + c*y*z^3 a square, where w is a positive integer and x,y,z are nonnegative integers.
The author has proved part (ii) of the conjecture in arXiv:1604.06723. - Zhi-Wei Sun, May 09 2016

			a(1) = 1 since 1 = 0^2 + 0^2 + 1^2 + 0^2 with 0 = 0, 1 > 0 and 0^4 + 8*1*0*(1^2+0^2) = 0^4.
a(2) = 1 since 2 = 1^2 + 0^2 + 1^2 + 0^2 with 1 > 0 and 0^4 + 8*1*0*(1^2+0^2) = 0^4.
a(3) = 1 since 3 = 1^2 + 1^2 + 1^2 + 0^2 with 1 = 1, 1 > 0 and 1^4 + 8*1*0*(1^2+0^2) = 1^4.
a(7) = 1 since 7 = 1^2 + 1^2 + 2^2 + 1^2 with 1 = 1, 3 > 1 and 1^4 + 8*2*1*(2^2+1^2) = 3^4.
a(31) = 1 since 31 = 5^2 + 1^2 + 2^2 + 1^2 with 5 > 1, 2 > 1 and 1^4 + 8*2*1*(2^2+1^2) = 3^4.
a(55) = 1 since 55 = 7^2 + 1^2 + 2^2 + 1^2 with 7 > 1, 2 > 1 and 1^4 + 8*2*1*(2^2+1^2) = 3^4.
a(71) = 1 since 71 = 3^2 + 1^2 + 6^2 + 5^2 with 3 > 1, 6 > 5 and 1^4 + 8*6*5*(6^2+5^2) = 11^4.
a(79) = 1 since 79 = 5^2 + 3^2 + 6^2 + 3^2 with 5 > 3, 6 > 3 and 3^4 + 8*6*3*(6^2+3^2) = 9^4.
a(151) = 1 since 151 = 5^2 + 3^2 + 9^2 + 6^2 with 5 > 3, 9 > 6 and 3^4 + 8*9*6*(9^2+6^2) = 15^4.
a(191) = 1 since 191 = 3^2 + 1^2 + 10^2 + 9^2 with 3 > 1, 10 > 9 and 1^4 + 8*10*9*(10^2+9^2) = 19^4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016.

Cf. A000118, A000290, A000583, A262357, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&QQ[x^4+8y*z*(y^2+z^2)],r=r+1],{z,0,(Sqrt[2n-1]-1)/2},{y,z+1,Sqrt[n-z^2]},{x,0,Sqrt[(n-y^2-z^2)/2]}];Print[n," ",r];Continue,{n,1,80}]

A260625 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+3y+13z)xy*z a square, where x is a positive integer, and y,z,w are nonnegative integers with y >= z.

Original entry on oeis.org

1, 2, 1, 1, 4, 4, 1, 2, 4, 5, 3, 1, 4, 7, 2, 1, 7, 6, 5, 6, 6, 5, 4, 4, 6, 11, 4, 3, 10, 7, 2, 2, 7, 7, 8, 4, 4, 10, 1, 5, 13, 7, 3, 5, 10, 6, 1, 1, 8, 13, 7, 5, 10, 13, 5, 7, 7, 6, 9, 3, 10, 13, 3, 1, 15, 13, 5, 10, 12, 8, 3, 6, 8, 16, 8, 8, 14, 8, 2, 6

Offset: 1

Zhi-Wei Sun, Apr 30 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 7, 39, 47, 95, 191, 239, 327, 439, 871, 1167, 1199, 1367, 1487, 1727, 1751, 2063, 2351, 2471, 4647, 4^k*m (k = 0,1,2,... and m = 1, 3).
(ii) 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+c*z)*x*y*z is a square, whenever (a,b,c) is among the triples (1,3,7), (1,5,7), (1,5,11), (1,13,23), (2,4,6), (2,4,8), (2,6,8), (2,8,26), (3,5,21), (3,7,15), (3,9,43), (3,9,69), (3,9,141), (3,21,27), (3,27,39), (3,33,45), (3,39,123), (6,8,12), (6,8,18), (6,8,22), (6,8,28), (6,12,48), (6,18,132), (6,24,34), (6,24,36), (6,42,72), (7,13,29), (7,19,23), (12,18,24), (12,18,30), (12,26,48), (13,15,21), (13,17,19), (13,33,39), (14,28,58), (15,45,51), (16,22,62), (18,22,24), (21,27,33), (21,27,57), (23,37,61), (24,54,66), (33,57,79), (38,48,66), (42,58,84), (46,92,118).
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.

			a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1 > 0 and (1+3*1+13*0)*1*1*0 =0^2.
a(4) = 1 since 4 = 2^2 + 0^2 + 0^2 + 0^2 with 2 > 0, 0 = 0 and (2+3*0+13*0)*2*0*0 = 0^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2 > 0, 1 = 1 and
(2+3*1+13*1)*2*1*1 = 6^2.
a(39) = 1 since 39 = 2^2 + 3^2 + 1^2 + 5^2 with 2 > 0, 3 > 1 and (2+3*3+13*1)*2*3*1 = 12^2.
a(47) = 1 since 47 = 2^2 + 3^2 + 3^2 + 5^2 with 2 > 0, 3 = 3 and (2+3*3+13*3)*2*3*3 = 30^2.
a(95) = 1 since 95 = 2^2 + 3^2 + 1^2 + 9^2 with 2 > 0, 3 > 1 and (2+3*3+13*1)*2*3*1 = 12^2.
a(191) = 1 since 191 = 2^2 + 3^2 + 3^2 + 13^2 with 2 > 0, 3 = 3 and (2+3*3+13*3)*2*3*3 = 30^2.
a(239) = 1 since 239 = 2^2 + 3^2 + 1^2 + 15^2 with 2 > 0, 3 > 1 and (2+3*3+13*1)*2*3*1 = 12^2.
a(327) = 1 since 327 = 11^2 + 3^2 + 1^2 + 14^2 with 11 > 0, 3 > 1 and (11+3*3+13*1)*11*3*1 = 33^2.
a(439) = 1 since 439 = 10^2 + 5^2 + 5^2 + 17^2 with 10 > 0, 5 = 5 and (10+3*5+13*5)*10*5*5 = 150^2.
a(871) = 1 since 871 = 21^2 + 15^2 + 3^2 + 14^2 with 21 > 0, 15 > 3 and (21+3*15+13*3)*21*15*3 = 315^2.
a(1167) = 1 since 1167 = 22^2 + 11^2 + 11^2 + 21^2 with 22 > 0, 11 = 11 and (22+3*11+13*11)*22*11*11 = 726^2.
a(1199) = 1 since 1199 = 14^2 + 21^2 + 21^2 + 11^2 with 14 > 0, 21 = 21 and (14+3*21+13*21)*14*21*21 = 1470^2.
a(1367) = 1 since 1367 = 14^2 + 21^2 + 21^2 + 17^2 with 14 > 0, 21 = 21 and (14+3*21+13*21)*14*21*21 = 1470^2.
a(1487) = 1 since 1487 = 9^2 + 29^2 + 6^2 + 23^2 with 9 > 0, 29 > 6 and (9+3*29+13*6)*9*29*6 = 522^2.
a(1727) = 1 since 1727 = 2^2 + 21^2 + 21^2 + 29^2 with 2 > 0, 21 = 21 and (2+3*21+13*21)*2*21*21 = 546^2.
a(1751) = 1 since 1751 = 9^2 + 17^2 + 15^2 + 34^2 with 9 > 0, 17 > 15 and (9+3*17+13*15)*9*17*15 = 765^2.
a(2063) = 1 since 2063 = 18^2 + 19^2 + 3^2 + 37^2 with 18 > 0, 19 > 3 and (18+3*19+13*3)*18*19*3 = 342^2.
a(2351) = 1 since 2351 = 15^2 + 35^2 + 15^2 + 26^2 with 15 > 0, 35 > 15 and (15+3*35+13*15)*15*35*15 = 1575^2.
a(2471) = 1 since 2471 = 1^2 + 18^2 + 11^2 + 45^2 with 1 > 0, 18 > 11 and (1+3*18+13*11)*1*18*11 = 198^2.
a(4647) = 1 since 4647 = 10^2 + 45^2 + 29^2 + 41^2 with 10 > 0, 45 > 29 and (10+3*45+13*29)*10*45*29 = 2610^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016.

Cf. A000118, A000290, A262357, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(x+3y+13z)x*y*z], r=r+1],{x,1,Sqrt[n]},{z,0,Sqrt[(n-x^2)/2]},{y,z,Sqrt[n-x^2-z^2]}];Print[n," ",r];Label[aa];Continue,{n,1,80}]

A272620 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w + x + y - z a square, where w is an integer and x,y,z are nonnegative integers with |w| <= x >= y <= z < x + y.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 3, 3, 2, 3, 1, 7, 1, 2, 3, 2, 1, 3, 3, 7, 2, 3, 1, 7, 1, 1, 4, 5, 3, 2, 1, 9, 2, 5, 3, 6, 5, 3, 3, 7, 2, 2, 5, 6, 3, 3, 5, 9, 4, 4, 4, 9, 4, 4, 5, 6, 6, 1, 6, 12, 2, 2, 7, 4, 4, 6, 5, 11, 7, 3, 5, 9, 4, 5

Offset: 1

Zhi-Wei Sun, May 03 2016

nonn

Conjecture: a(n) > 0 for all n > 0.
In contrast, the author has proved that any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that x + y + z is a square. See arXiv:1604.06723.
Yu-Chen Sun and the author proved in arXiv:1605.03074 that any nonnegative integer can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that w + x + y + z is a square. - Zhi-Wei Sun, May 10 2016

			a(1) = 1 since 1 = 0^2 + 1^2 + 0^2 + 0^2 with 0 < 1 > 0 = 0 < 1 + 0 and 0 + 1 + 0 - 0 = 1^2.
a(2) = 1 since 2 = (-1)^2 + 1^2 + 0^2 + 0^2 with 1 = 1 > 0 = 0 < 1 + 0 and -1 + 1 + 0 - 0 = 0^2.
a(3) = 1 since 3 = 0^2 + 1^2 + 1^2 + 1^2 with 0 < 1 = 1 = 1 < 1 + 1 and 0 + 1 + 1 - 1 = 1^2.
a(4) = 1 since 4 = (-1)^2 + 1^2 + 1^2 + 1^2 with 1 = 1 = 1 = 1 < 1 + 1 and -1 + 1 + 1 - 1 = 0^2.
a(6) = 1 since 6 = (-1)^2 + 2^2 + 0^2 + 1^2 with 1 < 2 > 0 < 1 < 2 + 0 and -1 + 2 + 0 - 1 = 0^2.
a(7) = 1 since 7 = (-1)^2 + 2^2 + 1^2 + 1^2 with 1 < 2 > 1 = 1 < 2 + 1 and -1 + 2 + 1 - 1 = 1^2.
a(9) = 1 since 9 = 0^2 + 2^2 + 1^2 + 2^2 with 0 < 2 > 1 < 2 < 2 + 1 and 0 + 2 + 1 - 2 = 1^2.
a(11) = 1 since 11 = (-1)^2 + 3^2 + 0^2 + 1^2 with 1 < 3 > 0 < 1 < 3 + 0 and -1 + 3 + 0 - 1 = 1^2.
a(12) = 1 since 12 = 1^2 + 3^2 + 1^2 + 1^2 with 1 < 3 > 1 = 1 < 3 + 1 and 1 + 3 + 1 - 1 = 2^2.
a(17) = 1 since 17 = 0^2 + 2^2 + 2^2 + 3^2 with 0 < 2 = 2 < 3 < 2 + 2 and 0 + 2 + 2 - 3 = 1^2.
a(19) = 1 since 19 = 0^2 + 3^2 + 1^2 + 3^2 with 0 < 3 > 1 < 3 < 3 + 1 and 0 + 3 + 1 - 3 = 1^2.
a(23) = 1 since 23 = (-1)^2 + 3^2 + 2^2 + 3^2 with 1 < 3 > 2 < 3 < 3 + 2 and -1 + 3 + 2 - 3 = 1^2.
a(29) = 1 since 29 = 0^2 + 3^2 + 2^2 + 4^2 with 0 < 3 > 2 < 4 < 3 + 2 and 0 + 3 + 2 - 4 = 1^2.
a(31) = 1 since 31 = (-2)^2 + 3^2 + 3^2 + 3^2 with 2 < 3 = 3 = 3 < 3 + 3 and -2 + 3 + 3 - 3 = 1^2.
a(37) = 1 since 37 = (-1)^2 + 4^2 + 2^2 + 4^2 with 1 < 4 > 2 < 4 < 4 + 2 and -1 + 4 + 2 - 4 = 1^2.
a(92) = 1 since 92 = 3^2 + 5^2 + 3^2 + 7^2 with 3 < 5 > 3 < 7 < 5 + 3 and 3 + 5 + 3 - 7 = 2^2.
a(284) = 1 since 284 = 3^2 + 9^2 + 5^2 + 13^2 with 3 < 9 > 5 < 13 < 9 + 5 and 3 + 9 + 5 - 13 = 2^2.
a(572) = 1 since 572 = 3^2 + 11^2 + 9^2 + 19^2 with 3 < 11 > 9 < 19 < 11 + 9 and 3 + 11 + 9 - 19 = 2^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Yu-Chen Sun and Zhi-Wei Sun, Two refinements of Lagrange's four-square theorem, arXiv:1605.03074 [math.NT], 2016.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351.

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

Rick L. Shepherd, May 27 2016: I checked all the statements in each example.

A261876 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (5x^2+7y^2+9z^2)y*z a square, where x,y,z,w are nonnegative integers with z > 0.

Original entry on oeis.org

1, 3, 2, 1, 4, 5, 1, 3, 5, 5, 4, 2, 4, 7, 2, 1, 9, 9, 4, 4, 7, 5, 1, 5, 6, 12, 7, 1, 10, 9, 2, 3, 10, 9, 7, 5, 4, 11, 3, 5, 14, 10, 4, 4, 10, 9, 3, 2, 8, 17, 10, 4, 11, 18, 6, 7, 9, 6, 11, 2, 10, 15, 4, 1, 15, 17, 4, 9, 13, 10

Offset: 1

Zhi-Wei Sun, May 01 2016

nonn

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, 7, 23, 647, 863).
(ii) For each triple (a,b,c) = (1,8,20), (3,5,15), (6,14,4), (7,29,5), (18,38,18), (39,81,51), (42,98,14), 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*(a*x^2+b*y^2+c*z^2) is a square.
For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.

			a(4) = 1 since 4 = 0^2 + 0^2 + 2^2 + 0^2 with 2 > 0 and (5*0^2+7*0^2+9*2^2)*0*2 = 0^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 > 0 and (5*2^2+7*1^2+9*1^2)*1*1 = 6^2.
a(23) = 1 since 23 = 2^2 + 1^2 + 3^2 + 3^2 with 3 > 0 and (5*2^2+7*1^2+9*3^2)*1*3 = 18^2.
a(647) = 1 since 647 = 13^2 + 1^2 + 6^2 + 21^2 with 6 > 0 and (5*13^2+7*1^2+9*6^2)*1*6 = 84^2.
a(863) = 1 since 863 = 1^2 + 23^2 + 18^2 + 3^2 with 18 > 0 and (5*1^2+7*23^2+9*18^2)*23*18 = 1656^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016.

Cf. A000118, A000290, A260625, A262357, A267121, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[y*z(5x^2+7y^2+9z^2)],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[n-1-x^2]},{z,1,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,70}]

cp's OEIS Frontend

A271824 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+2*y)^2 + 8*z^2 + 40*w^2 a square, where x is a positive integer and y,z,w are nonnegative integers.

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A262357 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w^2*x^2 + 5*x^2*y^2 + 80*y^2*z^2 + 20*z^2*w^2 a square, where w is a positive integer and x,y,z are nonnegative integers.

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A268507 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w > 0, w >= x <= y <= z such that x^2*y^2 + y^2*z^2 + z^2*x^2 is a square, where w,x,y,z are nonnegative integers.

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A269400 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 6*w^2*x^2 + 12*x^2*y^2 + 52*y^2*z^2 + 27*z^2*w^2 a square, where w,x,y are nonnegative integers and z is a positive integer.

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A272084 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 4*x^2 + 5*y^2 + 20*z*w a square, where x,y,z,w are nonnegative integers with z < w.

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A272332 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 36*x^2*y + 12*y^2*z + z^2*x a square, where w is a positive integer and x,y,z are nonnegative integers.

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A272351 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with x^4 + 8*y*z*(y^2+z^2) a fourth power, where w,x,y,z are nonnegative integers with w >= x and y > z.

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A260625 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+3*y+13*z)*x*y*z a square, where x is a positive integer, and y,z,w are nonnegative integers with y >= z.

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A271824 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+2y)^2 + 8z^2 + 40*w^2 a square, where x is a positive integer and y,z,w are nonnegative integers.

A262357 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w^2x^2 + 5x^2y^2 + 80y^2z^2 + 20z^2*w^2 a square, where w is a positive integer and x,y,z are nonnegative integers.

A268507 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w > 0, w >= x <= y <= z such that x^2y^2 + y^2z^2 + z^2*x^2 is a square, where w,x,y,z are nonnegative integers.

A269400 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 6w^2x^2 + 12x^2y^2 + 52y^2z^2 + 27z^2w^2 a square, where w,x,y are nonnegative integers and z is a positive integer.

A272084 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with 4x^2 + 5y^2 + 20zw a square, where x,y,z,w are nonnegative integers with z < w.

A272332 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 36x^2y + 12y^2z + z^2*x a square, where w is a positive integer and x,y,z are nonnegative integers.

A272351 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with x^4 + 8yz*(y^2+z^2) a fourth power, where w,x,y,z are nonnegative integers with w >= x and y > z.

A260625 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+3y+13z)xy*z a square, where x is a positive integer, and y,z,w are nonnegative integers with y >= z.

A261876 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (5x^2+7y^2+9z^2)y*z a square, where x,y,z,w are nonnegative integers with z > 0.