A300708 -id:A300708 - cp's OEIS frontend

A300666 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with nonnegative integers x,y,z,w and z <= w such that x or 2y is a square and x + 3y is also a square.

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

Offset: 0

Zhi-Wei Sun, Mar 10 2018

nonn

Conjecture 1: a(n) > 0 for all n = 0,1,2,....
Conjecture 2: Any nonnegative integer n not equal to 3 can be written as x^2 + y^2 + z^2 + w^2 with nonnegative integers x,y,z,w such that x or 2*y is a square and 3*x - y is also a square.
By the author's 2017 JNT paper, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with integers x,y,z,w such that x (or 2*x) is a square.
See also A281976, A300667, A300708 and A300712 for similar conjectures.
a(n) > 0 for all n = 0..10^8. - Zhi-Wei Sun, Oct 04 2020

			a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and 0 + 3*0 = 0^2.
a(23) = 1 since 23 = 3^2 + 2^2 + 1^2 + 3^2 with 2*2 = 2^2 and 3 + 3*2 = 3^2.
a(56) = 1 since 56 = 4^2 + 0^2 + 2^2 + 6^2 with 4 = 2^2 and 4 + 3*0 = 2^2.
a(140) = 1 since 140 = 10^2 + 2^2 + 0^2 + 6^2 with 2*2 = 2^2 and 10 + 3*2 = 4^2.
a(472) = 1 since 472 = 0^2 + 12^2 + 2^2 + 18^2 with 0 = 0^2 and 0 + 3*12 = 6^2.
a(959) = 1 since 959 = 9^2 + 9^2 + 11^2 + 26^2 with 9 = 3^2 and 9 + 3*9 = 6^2.
a(1839) = 1 since 1839 = 1^2 + 5^2 + 7^2 + 42^2 with 1 = 1^2 and 1 + 3*5 = 4^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Yue-Feng She and Hai-Liang Wu, Proof of a conjecture of Sun on sums of four squares, arXiv:2010.02067 [math.NT], 2020.
Yue-Feng She and Hai-Liang Wu, Sums of four squares with a certain restriction, Bull. of the Australian Math. Soc. (2021) First View, 1-10.
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.

Cf. A000118, A000290, A271518, A281976, A300667, A300708, A300712.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[m^2-3y]||SQ[2y])&&SQ[n-(m^2-3y)^2-y^2-z^2],r=r+1],{m,0,(10n)^(1/4)},{y,0,Min[m^2/3,Sqrt[n]]},{z,0,Sqrt[Max[0,(n-(m^2-3y)^2-y^2)/2]]}];tab=Append[tab,r],{n,0,80}];Print[tab]

A300667 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that 3x or y is a square and x + 2y is also a square.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Mar 10 2018

nonn

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 = 0, 3, 7, 11, 12, 15, 28, 39, 47, 60, 71, 92, 119, 172, 232, 253, 263, 316, 347, 515.
Conjecture 2: Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 3*x or y is a square and 2*x - y is also a square.
By the author's 2017 JNT paper, any nonnegative integer can be written as the sum of a fourth power and three squares.
See also A281976, A300666, A300708 and A300712 for similar conjectures.
a(n) > 0 for all n = 0..10^8. Also, Conjecture 2 holds for all n = 0..10^8. In a 2018 paper Y.-C. Sun and Z.-W. Sun proved that any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x + 2*y a square, where x,y,z,w are nonnegative integers. - Zhi-Wei Sun, Oct 04 2020

			a(12) = 1 since 12 = 0^2 + 2^2 + 2^2 + 2^2 with 3*0 = 0^2 and 0 + 2*2 = 2^2.
a(39) = 1 since 39 = 2^2 + 1^2 + 3^2 + 5^2 with 1 = 1^2 and 2 + 2*1 = 2^2.
a(172) = 1 since 172 = 7^2 + 1^2 + 1^2 + 11^2 with 1 = 1^2 and 7 + 2*1 = 3^2.
a(232) = 1 since 232 = 0^2 + 0^2 + 6^2 + 14^2 with 0 = 0^2 and 0 + 2*0 = 0^2.
a(253) = 1 since 253 = 8^2 + 4^2 + 2^2 + 13^2 with 4 = 2^2 and 8 + 2*4 = 4^2.
a(263) = 1 since 263 = 3^2 + 3^2 + 7^2 + 14^2 with 3*3 = 3^2 and 3 + 2*3 = 3^2.
a(515) = 1 since 515 = 1^2 + 0^2 + 15^2 + 17^2 with 0 = 0^2 and 1 + 2*0 = 1^2.

Yu-Chen Sun and Zhi-Wei Sun, Some variants of Lagrange's four squares theorem, Acta Arith. 183(2018), 339-356.

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], 2017-2018.
Yu-Chen Sun and Zhi-Wei Sun, Some variants of Lagrange's four squares theorem, arXiv:1605.03074 [math.NT], 2016-2018.

Cf. A000118, A000290, A271518, A281976, A300666, A300708, A300712.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[3(m^2-2y)]||SQ[y])&&SQ[n-(m^2-2y)^2-y^2-z^2],r=r+1],{m,0,(5n)^(1/4)},{y,0,Min[m^2/2,Sqrt[n]]},{z,0,Sqrt[Max[0,(n-(m^2-2y)^2-y^2)/2]]}];tab=Append[tab,r],{n,0,80}];Print[tab]

A300667(n)=sum(x=0,sqrtint(n),sum(y=0,sqrtint(n-x^2),if(issquare(x+2*y)&&(issquare(y)||issquare(3*x)),if(n>x^2+y^2,A000161(n-x^2-y^2),1)))) \\ M. F. Hasler, Mar 11 2018

A300712 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 3*x or y is a square and x - y is twice a square.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Mar 11 2018

nonn

Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 16^k*m with k = 0,1,2,... and m = 0, 1, 3, 7, 21, 24, 46, 79, 88, 94, 142, 151, 184, 190, 193, 280, 286, 1336.
By the author's 2017 JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 2*(x-y) (or x) is a square.
See also A281976, A300666, A300667 and A300708 for similar conjectures.
a(n) > 0 for all n = 0..10^8. - Zhi-Wei Sun, Oct 04 2020

			a(21) = 1 since 21 = 2^2 + 0^2 + 1^2 + 4^2 with 0 = 0^2 and 2 - 0 = 2*1^2.
a(79) = 1 since 79 = 3^2 + 3^2 + 5^2 + 6^2 with 3*3 = 3^2 and 3 - 3 = 2*0^2.
a(142) = 1 since 142 = 6^2 + 4^2 + 3^2 + 9^2 with 4 = 2^2 and 6 - 4 = 2*1^2.
a(190) = 1 since 190 = 3^2 + 1^2 + 6^2 + 12^2 with 1 = 1^2 and 3 - 1 = 2*1^2.
a(193) = 1 since 193 = 0^2 + 0^2 + 7^2 + 12^2 with 0 = 0^2 and 0 - 0 = 2*0^2.
a(280) = 1 since 280 = 12^2 + 10^2 + 0^2 + 6^2 with 3*12 = 6^2 and 12 - 10 = 2*1^2.
a(1336) = 1 since 1336 = 2^2 + 0^2 + 6^2 + 36^2 with 0 = 0^2 and 2 - 0 = 2*1^2.

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], 2017-2018.

Cf. A000118, A000290, A271518, A271775, A281976, A300666, A300667, A300708.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[3(2m^2+y)]||SQ[y])&&SQ[n-(2m^2+y)^2-y^2-z^2],r=r+1],{m,0,(n/4)^(1/4)},{y,0,Sqrt[(n-4m^4)/2]},{z,0,Sqrt[Max[0,(n-(2m^2+y)^2-y^2)/2]]}];tab=Append[tab,r],{n,0,80}];Print[tab]

A300751 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3y + 5z a positive square, where x,y,z,w are nonnegative integers such that 2*x or y or z is a square.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 11 2018

nonn

Conjecture: a(n) > 0 for all n = 1,2,3,..., and a(n) = 1 only for n = 16^k*m with k = 0,1,2,... and m = 0, 1, 4, 5, 6, 7, 8, 20, 24, 28, 31, 36, 43, 61, 71, 79, 100, 116, 157, 188, 200, 344, 351, 388, 632.
This is stronger than the author's 1-3-5 conjecture in A271518. See also A300752 for a similar conjecture stronger than the 1-3-5 conjecture.
a(n) > 0 for all n = 1..3*10^6. - Zhi-Wei Sun, Oct 06 2020

			a(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 2*0 = 0^2 and 0 + 3*2 + 5*2 = 4^2.
a(61) = 1 since 61 = 0^2 + 0^2 + 5^2 + 6^2 with 0 = 0^2 and 0 + 3*0 + 5*5 = 5^2.
a(79) = 1 since 79 = 5^2 + 2^2 + 1^2 + 7^2 with 1 = 1^2 and 5 + 3*2 + 5*1 = 4^2.
a(188) = 1 since 188 = 7^2 + 9^2 + 3^2 + 7^2 with 9 = 3^2 and 7 + 3*9 + 5*3 = 7^2.
a(200) = 0 since 200 = 6^2 + 10^2 + 0^2 + 8^2 with 0 = 0^2 and 6 + 3*10 + 5*0 = 6^2.
a(632) = 1 since 632 = 6^2 + 16^2 + 18^2 + 4^2 with 16 = 4^2 and 6 + 3*16 + 5*18 = 12^2.
a(808) = 3 since 808 = 8^2 + 2^2 + 26^2 + 8^2 = 8^2 + 22^2 + 14^2 + 8^2 = 18^2 + 12^2 + 18^2 + 4^2  with 2*8 = 4^2, 2*18 = 6^2 and 8 + 3*2 + 5*26 = 8 + 3*22 + 5*14 = 18 + 3*12 + 5*18 = 12^2.

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.

Cf. A000118, A000290, A271518, A281976, A282542, A300666, A300667, A300708, A300712, A300752.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[2(m^2-3y-5z)]||SQ[y]||SQ[z])&&SQ[n-(m^2-3y-5z)^2-y^2-z^2],r=r+1],{m,1,(35n)^(1/4)},{y,0,Min[m^2/3,Sqrt[n]]},{z,0,Min[(m^2-3y)/5,Sqrt[n-y^2]]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A300752 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3y + 5z a positive square, where x,y,z,w are nonnegative integers such that 3*x or y or z is a square.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 11 2018

nonn

Conjecture: a(n) > 0 for all n = 1,2,3,....
This is stronger than the author's 1-3-5 conjecture in A271518. See also A300751 for a similar conjecture stronger than the 1-3-5 conjecture.
In 2020, A. Machiavelo, R. Reis and N. Tsopanidis verified a(n) > 0 for n up to 1.05*10^11. - Zhi-Wei Sun, Oct 06 2020

			a(71) = 1 since 71 = 3^2 + 1^2 + 6^2 + 5^2 with 1 = 1^2 and 3 + 3*1 + 5*6 = 6^2.
a(248) = 1 since 248 = 10^2 + 2^2 + 0^2 + 12^2 with 0 = 0^2 and 10 + 3*2 + 5*0 = 4^2.
a(263) = 1 since 263 = 13^2 + 2^2 + 9^2 + 3^2 with 9 = 3^2 and 13 + 3*2 + 5*9 = 8^2.
a(808) = 1 since 808 = 12^2 + 14^2 + 18^2 + 12^2 with 3*12 = 6^2 and 12 + 3*14 + 5*18 = 12^2.
a(1288) = 1 since 1288 = 12^2 + 18^2 + 26^2 + 12^2 with 3*12 = 6^2 and 12 + 3*18 + 5*26 = 14^2.
a(3544) = 1 since 3544 = 14^2 + 34^2 + 16^2 + 44^2 with 16 = 4^2 and 14 + 3*34 + 5*16 = 14^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, Report on Zhi-Wei Sun's "1-3-5 conjecture" and some of its refinements, arXiv:2005.13526 [math.NT], 2020.
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.

Cf. A000118, A000290, A271518, A281976, A282542, A300666, A300667, A300708, A300712, A300751.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[3(m^2-3y-5z)]||SQ[y]||SQ[z])&&SQ[n-(m^2-3y-5z)^2-y^2-z^2],r=r+1],{m,1,(35n)^(1/4)},{y,0,Min[m^2/3,Sqrt[n]]},{z,0,Min[(m^2-3y)/5,Sqrt[n-y^2]]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A300791 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 for which x or y or z is a square and (12x)^2 + (15y)^2 + (20*z)^2 is also a square.

Original entry on oeis.org

1, 3, 1, 1, 6, 1, 1, 3, 2, 8, 2, 2, 7, 2, 2, 1, 8, 6, 2, 8, 1, 3, 1, 1, 9, 8, 4, 3, 7, 3, 3, 3, 6, 9, 4, 4, 7, 5, 1, 8, 8, 4, 3, 3, 11, 2, 1, 1, 4, 11, 3, 8, 8, 4, 4, 2, 3, 8, 4, 2, 8, 3, 4, 1, 15, 9, 3, 9, 3, 5, 2, 6, 10, 11, 5, 3, 5, 6, 2, 6

Offset: 1

Zhi-Wei Sun, Mar 12 2018

nonn

Conjecture: 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, 4, 6, 7, 21, 23, 24, 39, 47, 86, 95, 344, 651, 764.
By the author's 2017 JNT paper, each n = 0,1,2,... can be written as the sum of a fourth power and three squares.
See also A300792 for two similar conjectures.

			a(6) = 1 since 6 = 0^2 + 1^2 + 1^2 + 2^2 with 0 = 0^2 and (12*0)^2 + (15*1)^2 + (20*1)^2 = 25^2.
a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with 1 = 1^2 and (12*1)^2 + (15*2)^2 + (20*1)^2 = 38^2.
a(21) = 1 since 21 = 4^2 + 0^2 + 1^2 + 2^2 with 4 = 2^2 and (12*4)^2 + (15*0)^2 + (20*1)^2 = 52^2.
a(39) = 1 since 39 = 5^2 + 2^2 + 1^2 + 3^2 with 1 = 1^2 and (12*5)^2 + (15*2)^2 + (20*1)^2 = 70^2.
a(344) = 1 since 344 = 0^2 + 10^2 + 10^2 + 12^2 with 0 = 0^2 and (12*0)^2 + (15*10)^2 + (20*10)^2 = 250^2.
a(764) = 1 since 764 = 7^2 + 3^2 + 25^2 + 9^2 with 25 = 5^2 and (12*7)^2 + (15*3)^2 + (20*25)^2 = 509^2.
a(8312) = 2 since 8312 = 42^2 + 36^2 + 34^2 + 64^2 with 36 = 6^2 and (12*42)^2 + (15*36)^2 + (20*34)^2 = 1004^2, and 8312 = 66^2 + 16^2 + 44^2 + 42^2 with 16 = 4^2 and (12*66)^2 + (15*16)^2 + (20*44)^2 = 1208^2.

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.

Cf. A000118, A000290, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300792.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[x]||SQ[y]||SQ[z])&&SQ[(12x)^2+(15y)^2+(20z)^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]

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 9x^2 + 16y^2 + 24*z^2 is also a square.

Original entry on oeis.org

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

Zhi-Wei Sun, Mar 12 2018

nonn

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.

			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.

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.

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

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]

A300844 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x or 2y or z is a square and (12x)^2 + (21y)^2 + (28z)^2 is also a square.

Original entry on oeis.org

1, 4, 4, 2, 5, 6, 2, 2, 4, 5, 7, 1, 3, 7, 2, 3, 5, 7, 7, 2, 6, 1, 2, 2, 2, 11, 7, 3, 3, 8, 5, 1, 4, 5, 9, 4, 6, 8, 6, 4, 7, 9, 3, 3, 2, 9, 2, 1, 3, 6, 16, 5, 9, 7, 6, 5, 1, 5, 9, 4, 4, 7, 5, 5, 5, 17, 6, 4, 7, 3, 6, 3, 6, 11, 11, 4, 3, 1, 8, 2, 6

Offset: 0

Zhi-Wei Sun, Mar 13 2018

nonn

Conjecture 1: a(n) > 0 for all n = 0,1,2,....
Conjecture 2: Any positive integer can be written as x^2 + y^2 + z^2 + w^2 with w a positive integer and x,y,z nonnegative integers such that x or y or z is a square and 144*x^2 + 505*y^2 + 720*z^2 is also a square.
By the author's 2017 JNT paper, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x (or 2*x) is a square.
In 2016, the author conjectured in A271510 that each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y >= z such that (3*x)^2 + (4*y)^2 + (12*z)^2 is a square.
See also A300791 and A300792 for similar conjectures.

			a(11) = 1 since 11 = 0^2 + 1^2 + 1^2 + 3^2 with 0 = 0^2 and (12*0)^2 + (21*1)^2 + (28*1)^2 = 35^2.
a(56) = 1 since 56 = 4^2 + 6^2 + 2^2 + 0^2 with 4 = 2^2 and (12*4)^2 + (21*6)^2 + (28*2)^2 = 146^2.
a(77) = 1 since 77 = 4^2 + 0^2 + 5^2 + 6^2 with 4 = 2^2 and (12*4)^2 + (21*0)^2 + (28*5)^2 = 148^2.
a(184) = 1 since 184 = 12^2 + 2^2 + 0^2 + 6^2 with 0 = 0^2 and (12*12)^2 + (21*2)^2 + (28*0)^2 = 150^2.
a(599) = 1 since 599 = 21^2 + 11^2 + 1^2 + 6^2 with 1 = 1^2 and (12*21)^2 + (21*11)^2 + (28*1)^2 = 343^2.
a(7836) = 1 since 7836 = 38^2 + 18^2 + 68^2 + 38^2 with 2*18 = 6^2 and (12*38)^2 + (21*18)^2 + (28*68)^2 = 1994^2.
a(15096) = 1 since 15096 = 16^2 + 6^2 + 52^2 + 110^2 with 16 = 4^2 and (12*16)^2 + (21*6)^2 + (28*52)^2 = 1474^2.

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], 2017-2018.

Cf. A000118, A000290, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300791, A300792.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[x]||SQ[2y]||SQ[z])&&SQ[(12x)^2+(21y)^2+(28z)^2]&&SQ[n-x^2-y^2-z^2],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];tab=Append[tab,r],{n,0,80}]

A300908 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with (3x)^2 + (4y)^2 + (12z)^2 a square , where w is a positive integer and x,y,z are nonnegative integers such that z or 2z or 3*z is a square.

Original entry on oeis.org

1, 3, 1, 2, 6, 1, 2, 3, 2, 7, 2, 2, 7, 1, 5, 2, 7, 9, 3, 6, 2, 3, 4, 1, 9, 7, 3, 4, 5, 7, 2, 3, 5, 6, 3, 4, 7, 3, 8, 6, 7, 6, 4, 3, 7, 3, 2, 2, 4, 13, 5, 8, 6, 5, 3, 1, 8, 8, 3, 6, 5, 1, 11, 2, 16, 5, 4, 8, 1, 8, 2, 7, 11, 7, 5, 4, 2, 13, 2, 6

Offset: 1

Zhi-Wei Sun, Mar 15 2018

nonn

Conjecture 1: a(n) > 0 for all n > 0. Moreover, any positive integer can be written as x^2 + y^2 + z^2 + w^2 with (3*x)^2 + (4*y)^2 + (12*z)^2 a square, where w is a positive integer and x,y,z are nonnegative integers for which one of z, z/2, z/3 is a square and z/2 (or z/3) is a power of 4 (including 1).
Conjecture 2: Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with (3*x)^2 + (4*y)^2 + (12*z)^2 a square, where x,y,z,w are nonnegative integers for which x or z is a square or y = 2^k for some k = 0,1,2,....
Conjecture 3. Let a,b,c be positive integers with a <= b <= c and gcd(a,b,c) = 1. If each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that (a*x)^2 + (b*y)^2 + (c*z)^2 is a square, then (a,b,c) must be among the three triples (3,4,12), (12,15,20) and (12,21,28).
We have verified Conjectures 1 and 2 for n up to 5*10^5 and 10^6 respectively.
See also A300791 and A300844 for similar conjectures.

			a(6) = 1 since 6 = 1^2 + 1^2 + 0^2 + 2^2 with 0 = 0^2 and (3*1)^2 + (4*1)^2 + (12*0)^2 = 5^2.
a(14) = 1 since 14 = 3^2 + 0^2 + 1^2 + 2^2 with 1 = 1^2 and (3*3)^2 + (4*0)^2 + (12*1)^2 = 15^2.
a(69) = 1 since 69 = 0^2 + 8^2 + 2^2 + 1^2 with 2*2 = 2^2 and (3*0)^2 + (4*8)^2 + (12*2)^2 = 40^2.
a(671) = 1 since 671 = 18^2 + 17^2 + 3^2 + 7^2 with 3*3 = 3^2 and (3*18)^2 + (4*17)^2 + (12*3)^2 = 94^2.
a(1175) = 1 since 1175 = 30^2 + 5^2 + 9^2 + 13^2 with 9 = 3^2 and (3*30)^2 + (4*5)^2 + (12*9)^2 = 142^2.
a(12151) = 1 since 12151 = 50^2 + 71^2 + 49^2 + 47^2 with 49 = 7^2 and (3*50)^2 + (4*71)^2 + (12*49)^2 = 670^2.
a(16204) = 1 since 16204 = 90^2 + 90^2 + 0^2 + 2^2 with 0 = 0^2 and (3*90)^2 + (4*90)^2 + (12*0)^2 = 450^2.

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.

Cf. A000118, A000290, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300791, A300792, A300844.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[z]||SQ[2z]||SQ[3z], Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(3x)^2+(4y)^2+(12z)^2],r=r+1],{x,0,Sqrt[n-1-z^2]},{y,0,Sqrt[n-1-x^2-z^2]}]],{z,0,Sqrt[n-1]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A301303 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= 0 <= z <= w such that x^2 + 23y^2 = 2^km^3 for some k = 0,1,2 and m = 1,2,3,....

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 17 2018

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 2^k (k = 0,1,2,...), 3, 7, 115, 151, 219, 267, 1151, 1367.
We have verified a(n) > 0 for all n = 1..10^8.
See also A301304 and A301314 for similar conjectures.

			a(2) = 1 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 + 23*0 = 1^3.
a(4) = 1 since 4 = 2^2 + 0^2 + 0^2 + 0^2 with 2^2 + 23*0 = 2^2*1^3.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2^2 + 23*1^2 = 3^3.
a(48) = 2 since 48 = 4^2 + 0^2 + 4^2 + 4^2 = 6^2 + 2^2 + 2^2 + 2^2 with 4^2 + 23*0^2 = 2*2^3 and 6^2 + 23*2^2 = 2*4^3.
a(115) = 1 since 115 = 3^2 + 3^2 + 4^2 + 9^2 with 3^2 + 23*3^2 = 6^3.
a(267) = 1 since 267 = 3^2 + 1^2 + 1^2 + 16^2 with 3^2 + 23*1^2 = 2^2*2^3.
a(1151) = 1 since 1151 = 7^2 + 3^2 + 2^2 + 33^2 with 7^2 + 23*3^2 = 2^2*4^3.
a(1367) = 1 since 1367 = 17^2 + 5^2 + 18^2 + 27^2 with 17^2 + 23*5^2 = 2^2*6^3.

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.

Cf. A000118, A000290, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300791, A300792, A300844, A300908, A301304, A301314.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
QQ[n_]:=n>0&&(CQ[n]||CQ[n/2]||CQ[n/4]);
tab={};Do[r=0;Do[If[QQ[x^2+23y^2],Do[If[SQ[n-x^2-y^2-z^2],r=r+1],{z,0,Sqrt[(n-x^2-y^2)/2]}]],{y,0,Sqrt[n/2]},{x,y,Sqrt[n-y^2]}];tab=Append[tab,r],{n,1,80}];Print[tab]

cp's OEIS Frontend

A300666 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with nonnegative integers x,y,z,w and z <= w such that x or 2*y is a square and x + 3*y is also a square.

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A300667 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that 3*x or y is a square and x + 2*y is also a square.

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A300712 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 3*x or y is a square and x - y is twice a square.

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A300751 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3*y + 5*z a positive square, where x,y,z,w are nonnegative integers such that 2*x or y or z is a square.

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A300752 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3*y + 5*z a positive square, where x,y,z,w are nonnegative integers such that 3*x or y or z is a square.

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A300791 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 for which x or y or z is a square and (12*x)^2 + (15*y)^2 + (20*z)^2 is also a square.

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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.

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A300844 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x or 2*y or z is a square and (12*x)^2 + (21*y)^2 + (28*z)^2 is also a square.

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A300666 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with nonnegative integers x,y,z,w and z <= w such that x or 2y is a square and x + 3y is also a square.

A300667 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that 3x or y is a square and x + 2y is also a square.

A300751 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3y + 5z a positive square, where x,y,z,w are nonnegative integers such that 2*x or y or z is a square.

A300752 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3y + 5z a positive square, where x,y,z,w are nonnegative integers such that 3*x or y or z is a square.

A300791 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 for which x or y or z is a square and (12x)^2 + (15y)^2 + (20*z)^2 is also a square.

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 9x^2 + 16y^2 + 24*z^2 is also a square.

A300844 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x or 2y or z is a square and (12x)^2 + (21y)^2 + (28z)^2 is also a square.

A300908 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with (3x)^2 + (4y)^2 + (12z)^2 a square , where w is a positive integer and x,y,z are nonnegative integers such that z or 2z or 3*z is a square.

A301303 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x >= y >= 0 <= z <= w such that x^2 + 23y^2 = 2^km^3 for some k = 0,1,2 and m = 1,2,3,....