A282494 -id:A282494 - cp's OEIS frontend

A282495 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 228yz + 60*z^2 a square, where x,y,z are nonnegative integers and w is a positive integer.

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

Offset: 1

Zhi-Wei Sun, Feb 16 2017

nonn

Conjecture: a(n) > 0 for all n > 0.
By the linked JNT paper, any nonnegative integer can be expressed as the sum of a fourth power and three squares, and 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 z*(y-z) = 0. Whether z = 0 or y = z, the number y^2 + 228*y*z + 60*z^2 is definitely a square.
See also A282463 and A282494 for similar conjectures.

			a(7) = 1 since 7 = 1^4 + 1^2 + 1^2 + 2^2 with 1^2 + 228*1*1 + 60*1^2 = 17^2.
a(8) = 1 since 8 = 0^4 + 2^2 + 0^2 + 2^2 with 2^2 + 228*2*0 + 60*0^2 = 2^2.
a(15) = 1 since 15 = 1^4 + 1^2 + 3^2 + 2^2 with 1^2 + 228*1*3 + 60*3^2 = 35^2.
a(23) = 1 since 23 = 1^4 + 3^2 + 3^2 + 2^2 with 3^2 + 228*3*3 + 60*3^2 = 51^2.
a(71) = 1 since 71 = 1^4 + 5^2 + 6^2 + 3^2 with 5^2 + 228*5*6 + 60*6^2 = 95^2.
a(159) = 1 since 159 = 3^4 + 7^2 + 2^2 + 5^2 with 7^2 + 228*7*2 + 60*2^2 = 59^2.
a(623) = 1 since 623 = 3^4 + 1^2 + 10^2 + 21^2 with 1^2 + 228*1*10 + 60*10^2 = 91^2.
a(879) = 1 since 879 = 5^4 + 5^2 + 15^2 + 2^2 with 5^2 + 228*5*15 + 60*15^2 = 175^2.
a(1423) = 1 since 1423 = 1^4 + 7^2 + 2^2 + 37^2 with 7^2 + 228*7*2 + 60*2^2 = 59^2.
a(3768) = 1 since 3768 = 0^4 + 2^2 + 20^2 + 58^2 with 2^2 + 228*2*20 + 60*20^2 = 182^2.

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, A271518, A281976, A281977, A282013, A282014, A282463, A282494.

Q[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[y^2+228*y*z+60*z^2],r=r+1],{x,0,(n-1)^(1/4)},{y,0,Sqrt[n-1-x^4]},{z,0,Sqrt[n-1-x^4-y^2]}];Print[n," ",r];Continue,{n,1,80}]

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.

Original entry on oeis.org

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}]

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}]

A282933 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that 8y^2 - 8yz + 9z^2 is a square.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 25 2017

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 16^k*m (k = 0,1,2,... and m = 1, 8, 15, 23, 39, 47, 71, 93, 239, 287, 311, 319, 383, 391, 591, 632, 1663, 2639, 5591, 6236).
(ii) Each n = 0,1,2,... 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 (6,21,19), (15,33,22), (16,54,39),(18,51,34), (19,53,34), (21,42,22), (22,69,51).
By the linked JNT paper, each n = 0,1,2,... is the sum of a fourth power and three squares, and we can also write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y*(y-z) = 0. Whether y = 0 or y = z, the number 8*y^2 - 8*y*z + 9*z^2 is definitely a square.
First occurrence of k: 1, 2, 5, 6, 10, 18, 26, 25, 41, 85, 81, 101, 105, 90, 201, 146, 321, 341, 261, 325, 297, 370, 585, 306, 906, ..., . Robert G. Wilson v, Feb 25 2017

			a(8) = 1 since 8 = 0^4 + 0^2 + 2^2 + 2^2 with 8*0^2 - 8*0*2 + 9*2^2 = 6^2.
a(15) = 1 since 15 = 1^4 + 2^2 + 1^2 + 3^2 with 8*2^2 - 8*2*1 + 9*1^2 = 5^2.
a(23) = 1 since 23 = 1^4 + 3^2 + 3^2 + 2^2 with 8*3^2 - 8*3*3 + 9*3^2 = 9^2.
a(591) = 1 since 591 = 3^4 + 5^2 + 1^2 + 22^2 with 8*5^2 - 8*5*1 + 9*1^2 = 13^2.
a(632) = 1 since 632 = 4^4 + 12^2 + 6^2 + 14^2 with 8*12^2 - 8*12*6 + 9*6^2 = 30^2.
a(1663) = 1 since 1663 = 3^4 + 27^2 + 23^2 + 18^2 with 8*27^2 - 8*27*23 + 9*23^2 = 75^2.
a(2639) = 1 since 2639 = 7^4 + 15^2 + 3^2 + 2^2 with 8*15^2 - 8*15*3 + 9*3^2 = 39^2.
a(5591) = 1 since 5591 = 5^4 + 6^2 + 21^2 + 67^2 with 8*6^2 - 8*6*21 + 9*21^2 = 57^2.
a(6236) = 1 since 6236 = 1^4 + 45^2 + 31^2 + 57^2 with 8*45^2 - 8*45*31 + 9*31^2 = 117^2.

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, A281977, A282013, A282014, A282494, A282495.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[8y^2-8*y*z+9z^2],r=r+1],{x,0,(n-1)^(1/4)},{y,0,Sqrt[n-1-x^4]},{z,0,Sqrt[n-1-x^4-y^2]}];Print[n," ",r];Continue,{n,1,80}]

cp's OEIS Frontend

A282495 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 228*y*z + 60*z^2 a square, where x,y,z are nonnegative integers and w is a positive integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A282933 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that 8*y^2 - 8*y*z + 9*z^2 is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A282495 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 228yz + 60*z^2 a square, where x,y,z are nonnegative integers and w is a positive integer.

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.

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.

A282933 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that 8y^2 - 8yz + 9z^2 is a square.