A282545 -id:A282545 - cp's OEIS frontend

A282561 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2x - y and 4z^2 + 724zw + w^2 are squares.

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

Offset: 0

Zhi-Wei Sun, Feb 18 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 7, 19, 67, 191, 235, 265, 347, 888, 2559).
By the linked JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z*(z-w) = 0. Whether z = 0 or z = w, the number 4*z^2 + 724*z*w + w^2 is definitely a square.
See also A282562 for a similar conjecture.

			 a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with 2*1 - 2 = 0^2 and 4*1^2 + 724*1*1 + 1^2 = 27^2.
a(19) = 1 since 19 = 1^2 + 1^2 + 1^2 + 4^2 with 2*1 - 1 = 1^2 and 4*1^2 + 724*1*4 + 4^2 = 54^2.
a(67) = 1 since 67 = 4^2 + 7^2 + 1^2 + 1^2 with 2*4 - 7 = 1^2 and 4*1^2 + 724*1*1 + 1^2 = 27^2.
a(191) = 1 since 191 = 9^2 + 2^2 + 5^2 + 9^2 with 2*9 - 2 = 4^2 and 4*5^2 + 724*5*9 + 9^2 = 181^2.
a(235) = 1 since 235 = 7^2 + 13^2 + 1^2 + 4^2 with 2*7 - 13 = 1^2 and 4*1^2 + 724*1*4 + 4^2 = 54^2.
a(265) = 1 since 265 = 4^2 + 7^2 + 10^2 + 10^2 with 2*4 - 7 = 1 and 4*10^2 + 724*10*10 + 10^2 = 270^2.
a(347) = 1 since 347 = 8^2 + 7^2 + 15^2 + 3^2 with 2*8 - 7 = 3^2 and 4*15^2 + 724*15*3 + 3^2 = 183^2.
a(888) = 1 since 888 = 14^2 + 12^2 + 8^2 + 22^2 with 2*14 - 12 = 4^2 and 4*8^2 + 724*8*22 + 22^2 = 358^2.
a(2559) = 1 since 2559 = 26^2 + 3^2 + 5^2 + 43^2 with 2*26 - 3 = 7^2 and 4*5^2 + 724*5*43 + 43^2 = 397^2.

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.

Cf. A000118, A000290, A271518, A281976, A282463, A282494, A282495, A282545, A282562.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[2x-y],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[4z^2+724z*Sqrt[n-x^2-y^2-z^2]+(n-x^2-y^2-z^2)],r=r+1],{z,0,Sqrt[n-x^2-y^2]}]],{y,0,Sqrt[4n/5]},{x,Ceiling[y/2],Sqrt[n-y^2]}];Print[n," ",r];Continue,{n,0,80}]

A282562 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2x - y and 9z^2 + 666zw + w^2 are squares.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 5, 2, 3, 5, 2, 2, 1, 5, 6, 2, 2, 5, 5, 1, 5, 7, 8, 1, 3, 4, 2, 4, 3, 9, 7, 5, 2, 5, 5, 4, 8, 3, 9, 3, 4, 4, 5, 3, 1, 7, 6, 3, 2, 13, 7, 5, 5, 5, 8, 3, 2, 3, 5, 2, 1, 6, 10, 6, 4, 8, 9, 1, 7, 8, 11, 3, 3, 6, 6, 5, 3, 11, 4, 4, 4

Offset: 0

Zhi-Wei Sun, Feb 18 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 12, 19, 23, 44, 60, 67, 139, 140, 248, 264, 347, 427, 499, 636, 1388, 1867, 1964, 4843).
By the linked JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z*(z-w) = 0. Whether z = 0 or z = w, the number 9*z^2 + 666*z*w + w^2 is definitely a square.
See also A282561 for a similar conjecture.

			a(12) = 1 since 12 = 2^2 + 0^2 + 2^2 + 2^2 with 2*2 - 0 = 2^2 and 9*2^2 + 666*2*2 + 2^2 = 52^2.
a(19) = 1 since 19 = 1^2 + 1^2 + 4^2 + 1^2 with 2*1 - 1 = 1^2 and 9*4^2 + 666*4*1 + 1^2 = 53^2.
a(44) = 1 since 44 = 5^2 + 1^2 + 3^2 + 3^2 with 2*5 - 1 = 3^2 and 9*3^2 + 666*3*3 + 3^2 = 78^2.
a(60) = 1 since 60 = 3^2 + 5^2 + 1^2 + 5^2 with 2*3 - 5 = 1^2 and 9*1^2 + 666*1*5 + 5^2 = 58^2.
a(67) = 1 since 67 = 4^2 + 7^2 + 1^2 + 1^2 with 2*4 - 7 = 1^2 and 9*1^2 + 666*1*1 + 1^2 = 26^2.
a(139) = 1 since 139 = 8^2 + 7^2 + 1^2 + 5^2 with 2*8 - 7 = 3^2 and 9*1^2 + 666*1*5 + 5^2 = 58^2.
a(140) = 1 since 140 = 3^2 + 5^2 + 5^2 + 9^2 with 2*3 - 5 = 1^2 and 9*5^2 + 666*5*9 + 9^2 = 174^2.
a(264) = 1 since 264 = 8^2 + 0^2 + 10^2 + 10^2 with 2*8 - 0 = 4^2 and 9*10^2 + 666*10*10 + 10^2 = 260^2.
a(499) = 1 since 499 = 7^2 + 5^2 + 20^2 + 5^2 with 2*7 - 5 = 3^2 and 9*20^2 + 666*20*5 + 5^2 = 265^2.
a(1388) = 1 since 1388 = 15^2 + 21^2 + 19^2 + 19^2 with 2*15 - 21 = 3^2 and 9*19^2 + 666*19*19 + 19^2 = 494^2.
a(1867) = 1 since 1867 = 16^2 + 31^2 + 5^2 + 25^2 with 2*16 - 31 = 1^2 and 9*5^2 + 666*5*25 + 25^2 = 290^2.
a(4843) = 1 since 4843 = 11^2 + 13^2 + 52^2 + 43^2 with 2*11 - 13 = 3^2 and 9*52^2 + 666*52*43 + 43^2 = 1231^2.

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.

Cf. A000118, A000290, A271518, A281939, A281976, A282463, A282494, A282495, A282545, A282561.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[2x-y],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[9z^2+666z*Sqrt[n-x^2-y^2-z^2]+(n-x^2-y^2-z^2)],r=r+1],{z,0,Sqrt[n-x^2-y^2]}]],{y,0,Sqrt[4n/5]},{x,Ceiling[y/2],Sqrt[n-y^2]}];Print[n," ",r];Continue,{n,0,80}]

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A282561 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2x - y and 4z^2 + 724zw + w^2 are squares.

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A282562 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2x - y and 9z^2 + 666zw + w^2 are squares.

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cp's OEIS Frontend

A282561 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2*x - y and 4*z^2 + 724*z*w + w^2 are squares.

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A282562 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2*x - y and 9*z^2 + 666*z*w + w^2 are squares.

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A282561 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2x - y and 4z^2 + 724zw + w^2 are squares.

A282562 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2x - y and 9z^2 + 666zw + w^2 are squares.