A282542 -id:A282542 - cp's OEIS frontend

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.

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]

A282545 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 64z^2 + 1024y*z a square, where x,y,z,w are nonnegative integers with y > 0.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 18 2017

nonn

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) Any nonnegative integer can be written as x^4 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that a*y^2 + b*y*z + c*z^2 is a square, whenever (a,b,c) is among the ordered triples (1,484,44), (1,666,9), (16,1336,169), (25,900,36).
(iii) For each c = 1, 49, any nonnegative integer can be written as x^4 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 120*(x^2+y)*z + c*z^2 is a square.
By the linked JNT paper, each n = 0,1,2,... can be expressed as the sum of a fourth power and three squares.
See also A282463, A282494 and A282495 for similar conjectures.

			a(1) = 1 since 1 = 0^4 + 1^2 + 0^2 + 0^2 with 1^2 + 64*0^2 + 1024*1*0 = 1^2.
a(31) = 1 since 31 = 1^4 + 2^2 + 1^2 + 5^2 with 2^2 + 64*1^2 + 1024*2*1 = 46^2.
a(47) = 1 since 47 = 1^4 + 6^2 + 3^2 + 1^2 with 6^2 + 64*3^2 + 1024*6*3 = 138^2.
a(79) = 1 since 79 = 1^4 + 7^2 + 2^2 + 5^2 with 7^2 + 64*2^2 + 1024*7*2 = 121^2.
a(156) = 1 since 156 = 3^4 + 5^2 + 5^2 + 5^2 with 5^2 + 64*5^2 + 1024*5*5 = 165^2.
a(184) = 1 since 184 = 0^4 + 12^2 + 6^2 + 2^2 with 12^2 + 64*6^2 + 1024*12*6 = 276^2.
a(316) = 1 since 316 = 2^4 + 10^2 + 10^2 + 10^2 with 10^2 + 64*10^2 + 1024*10*10 = 330^2.
a(380) = 1 since 380 = 1^4 + 3^2 + 3^2 + 19^2 with 3^2 + 64*3^2 + 1024*3*3 = 99^2.
a(2383) = 1 since 2383 = 3^4 + 22^2 + 33^2 + 27^2 with 22^2 + 64*33^2 + 1024*22*33 = 902^2.

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.

Cf. A000118, A000290, A000583, A270969, A271518, A281976, A282463, A282494, A282495, A282542.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[y^2+1024y*z+64z^2],r=r+1],{x,0,(n-1)^(1/4)},{y,1,Sqrt[n-x^4]},{z,0,Sqrt[n-x^4-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A300139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 4x - 3y a square, where x,y,z,w are nonnegative integers with z <= w such that 10*x or y is a square.

Original entry on oeis.org

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

Offset: 0

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 = 0, 7, 8, 13, 22, 23, 24, 31, 44, 56, 71, 79, 88, 109, 120, 152, 184, 472, 1912, 6008, 9080.
Conjecture 2: Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with 3*x - y twice a square, where x,y,z,w are nonnegative integers such that 5*x or y is 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, A300667, 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. - Zhi-Wei Sun, Oct 05 2020

			a(22) = 1 since 22 = 1^2 + 1^2 + 2^2 + 4^2 with 1 = 1^2 and 4*1 - 3*1 = 1^2.
a(23) = 1 since 23 = 3^2 + 1^2 + 2^2 + 3^2 with 1 = 1^2 and 4*3 - 3*1 = 3^2.
a(109) = 1 since 109 = 0^2 + 0^2 + 3^2 + 10^2 with 0 = 0^2 and 4*0 - 3*0 = 0^2.
a(184) = 1 since 184 = 10^2 + 8^2 + 2^2 + 4^2 with 10*10 = 10^2 and 4*10 - 3*8 = 4^2.
a(6008) = 1 since 6008 = 12^2 + 16^2 + 42^2 + 62^2 with 16 = 4^2 and 4*12 - 3*16 = 0^2.
a(9080) = 1 since 9080 = 10^2 + 12^2 + 0^2 + 94^2 with 10*10 = 10^2 and 4*10 - 3*12 = 2^2.

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, A281976, A282542, A300666, A300667, A300219, A300708, A300712, A300751, A300752.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[Mod[m^2+3y,4]==0&&(SQ[10(m^2+3y)/4]||SQ[y]), Do[If[SQ[n-((m^2+3y)/4)^2-y^2-z^2],r=r+1],{z,0,Sqrt[Max[0,(n-((m^2+3y)/4)^2-y^2)/2]]}]],{m,0,2n^(1/4)},{y,0,4/5*Sqrt[n-m^4/16]}];tab=Append[tab,r],{n,0,80}];Print[tab]

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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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A282545 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 64*z^2 + 1024*y*z a square, where x,y,z,w are nonnegative integers with y > 0.

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A300139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 4*x - 3*y a square, where x,y,z,w are nonnegative integers with z <= w such that 10*x or y is a square.

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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.

A282545 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 64z^2 + 1024y*z a square, where x,y,z,w are nonnegative integers with y > 0.

A300139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 4x - 3y a square, where x,y,z,w are nonnegative integers with z <= w such that 10*x or y is a square.