A270969 -id:A270969 - cp's OEIS frontend

A282463 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers, x == y (mod 2) and z <= w such that both x and x^2 + 62xy + y^2 are squares.

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

Offset: 0

Zhi-Wei Sun, Feb 16 2017

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 3, 43, 723, 1723, 3571, 3911 and 16^k*m (k = 0,1,2,... and m = 7, 23, 31, 71, 79, 143, 303, 1591).
(ii) Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and a*x^2 + b*x*y + c*y^2 are squares, whenever (a,b,c) is among the ordered triples (84,84,1), (16,144,9), (153,36,100), (177,214,9), (249,114,121).
The author has proved that any nonnegative integer can be expressed as the sum of a fourth power and three squares.

			a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 == 1 (mod 2), 1 = 1^2 and 1^2 + 62*1*1 + 1^2 = 8^2.
a(23) = 1 since 23 = 1^2 + 3^2 + 2^2 + 3^2 with 1 == 1 (mod 3), 1 = 1^2 and 1^2 + 62*1*3 + 3^2 = 14^2.
a(30) = 2 since 30 = 0^2 + 2^2 + 1^2 + 5^2 with 0 == 2 (mod 3), 0 = 0^2 and 0^2 + 62*0*2 + 2^2 = 2^2, and 30 = 1^2 + 3^2 + 2^2 + 4^2 with 1 == 3 (mod 2), 1 = 1^2 and 1^2 + 62*1*3 + 3^2 = 14^2.
a(79) = 1 since 79 = 1^2 + 7^2 + 2^2 + 5^2 with 1 == 7 (mod 2), 1 = 1^2 and 1^2 + 62*1*7 + 7^2 = 22^2.
a(143) = 1 since 143 = 9^2 + 3^2 + 2^2 + 7^2 with 9 == 3 (mod 2), 9 = 3^2 and 9^2 + 62*9*3 + 3^2 = 42^2.
a(303) = 1 since 303 = 1^2 + 3^2 + 2^2 + 17^2 with 1 == 3 (mod 2), 1 = 1^2 and 1^2 + 62*1*3 + 3^2 = 14^2.
a(723) = 1 since 723 = 1^2 + 7^2 + 12^2 + 23^2 with 1 == 7 (mod 2), 1 = 1^2 and 1^2 + 62*1*7 + 7^2 = 22^2.
a(1591) = 1 since 1591 = 9^2 + 9^2 + 23^2 + 30^2 with 9 == 9 (mod 2), 9 = 3^2 and 9^2 + 62*9*9 + 9^2 = 72^2.
a(1723) = 1 since 1723 = 1^2 + 1^2 + 11^2 + 40^2 with 1 == 1 (mod 2), 1 = 1^2 and 1^2 + 62*1*1 + 1^2 = 8^2.
a(3571) = 1 since 3571 = 9^2 + 3^2 + 0^2 + 59^2 with 9 == 3 (mod 2), 9 = 3^2 and 9^2 + 62*9*3 + 3^2 = 42^2.
a(3911) = 1 since 9^2 + 3^2 + 10^2 + 61^2 with 9 == 3 (mod 2), 9 = 3^2 and 9^2 + 62*9*3 + 3^2 = 42^2.

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, A270969, A271518, A281976, A281977, A282013, A282014.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&Mod[x-y,2]==0&&SQ[x^4+62*x^2*y+y^2],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[(n-x^4-y^2)/2]}];Print[n," ",r];Continue,{n,0,80}]

A273458 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x-y+z+w a nonnegative cube, where x,y,z,w are integers with x >= y >= 0 and x >= |z| <= |w|.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 1, 5, 4, 3, 2, 1, 4, 3, 3, 6, 3, 2, 5, 3, 9, 3, 1, 1, 7, 5, 3, 7, 10, 4, 6, 2, 10, 2, 6, 2, 12, 7, 2, 5, 9, 3, 3, 6, 13, 3, 8, 3, 18, 3, 8, 5, 7, 3, 3, 5, 13, 8, 5, 3, 19, 4, 7, 7, 16, 1, 11, 5, 14, 7, 2, 3, 12, 5, 4

Offset: 0

Zhi-Wei Sun, May 22 2016

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,....
In the latest version of arXiv:1605.03074, the authors showed that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + y + z + w is a cube (or a square).
For more conjectural refinements of Lagrange's four-square theorem, see the author's preprint arXiv:1604.06723.

			a(12) = 1 since 12 = 3^2 + 1^2 + (-1)^2 + (-1)^2 with 3 - 1 + (-1) + (-1) = 0^3.
a(17) = 1 since 17 = 2^2 + 0^2 + 2^2 + (-3)^2 with 2 - 0 + 2 + (-3) = 1^3.
a(28) = 1 since 28 = 3^2 + 1^2 + 3^2 + 3^2 with 3 - 1 + 3 + 3 = 2^3.
a(29) = 1 since 29 = 3^2 + 0^2 + 2^2 + (-4)^2 with 3 - 0 + 2 + (-4) = 1^3.
a(71) = 1 since 71 = 5^2 + 1^2 + 3^2 + (-6)^2 with 5 - 1 + 3 + (-6) = 1^3.
a(149) = 1 since 149 = 8^2 + 0^2 + 2^2 + (-9)^2 with 8 - 0 + 2 + (-9) = 1^3.
a(188) = 1 since 188 = 13^2 + 3^2 + 1^2 + (-3)^2 with 13 - 3 + 1 + (-3) = 2^3.
a(284) = 1 since 284 = 15^2 + 5^2 + 3^2 + (-5)^2 with 15 - 5 + 3 + (-5) = 2^3.

Zhi-Wei Sun, Table of n, a(n) for n = 0..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.

Cf. A000118, A000290, A000578, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273429, A273432, A273568.

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

A273568 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w + x + 2y - 4z twice a nonnegative cube, where w is an integer and x,y,z are nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, May 25 2016

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,....

For more conjectural refinements of Lagrange's four-square theorem, see the author's preprint arXiv:1604.06723.

			a(1) = 1 since 1 = 0^2 + 0^2 + 1^2 + 0^2 with 0 + 0 + 2*1 - 4*0 = 2*1^3.
a(3) = 1 since 3 = (-1)^2 + 1^2 + 1^2 + 0^2 with (-1) + 1 + 2*1 - 4*0 = 2*1^3.
a(13) = 1 since 13 = (-2)^2 + 2^2 + 2^2 + 1^2 with (-2) + 2 + 2*2 - 4*1 = 2*0^3.
a(16) = 1 since 16 = 2^2 + 2^2 + 2^2 + 2^2 with 2 + 2 + 2*2 - 4*2 = 2*0^3.
a(26) = 1 since 26 = 3^2 + 3^2 + 2^2 + 2^2 with 3 + 3 + 2*2 - 4*2 = 2*1^3.
a(32) = 1 since 32 = (-4)^2 + 4^2 + 0^2 + 0^2 with (-4) + 4 + 2*0 - 4*0 = 2*0^3.
a(40) = 1 since 40 = (-2)^2 + 4^2 + 4^2 + 2^2 with (-2) + 4 + 2*4 - 4*2 = 2*1^3.
a(218) = 1 since 218 = (-6)^2 + 6^2 + 11^2 + 5^2 with (-6) + 6 + 2*11 - 4*5 = 2*1^3.
a(416) = 1 since 416 = (-4)^2 + 20^2 + 0^2 + 0^2 with (-4) + 20 + 2*0 - 4*0 = 2*2^3.
a(544) = 1 since 544 = (-4)^2 + 20^2 + 8^2 + 8^2 with (-4) + 20 + 2*8 - 4*8 = 2*0^3.
a(800) = 1 since 800 = (-20)^2 + 20^2 + 0^2 + 0^2 with (-20) + 20 + 2*0 - 4*0 = 2*0^3.
a(1184) = 1 since 1184 = (-28)^2 + 12^2 + 16^2 + 0^2 with (-28) + 12 + 2*16 - 4*0 = 2*2^3.
a(2080) = 1 since 2080 = (-20)^2 + 20^2 + 32^2 + 16^2 with (-20) + 20 + 2*32 - 4*16 = 2*0^3.
a(6304) = 1 since 6304 = (-36)^2 + 36^2 + 56^2 + 24^2 with (-36) + 36 + 2*56 - 4*24 = 2*2^3.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A000578, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273429, A273432, A273458.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
CQ[n_]:=CQ[n]=n>=0&&IntegerQ[n^(1/3)]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&CQ[(x+2y-4z+(-1)^k*Sqrt[n-x^2-y^2-z^2])/2],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]},{k,0,Min[1,n-x^2-y^2-z^2]}];Print[n," ",r];Continue,{n,0,70}]

A281975 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 both x and |x-y| are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 03 2017

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
(ii) Each nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that |x-y| and 2*(y-z) (or 2*(z-y)) are both squares.
(iii) For each ordered pair (a,b) = (2,1), (3,1), (9,5), (14,10), any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x and |a*x-b*y| are both squares.
The author has proved 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 such that x (or x-y, or 2(x-y)) is a square.
See also A281976 and A281977 for similar conjectures.

			a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and |0-0| = 0^2.
a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 = 1^2 and |1-1| = 0^2.
a(44) = 1 since 44 = 1^2 + 5^2 + 3^2 + 3^2 with 1 = 1^2 and |1-5| = 2^2.
a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 = 1^2 and |1-1| = 0^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 = 1^2 and |1-5| = 2^2.
a(95) = 1 since 95 = 1^2 + 2^2 + 3^2 + 9^2 with 1 = 1^2 and |1-2| = 1^2.
a(140) = 1 since 140 = 9^2 + 5^2 + 3^2 + 5^2 with 9 = 3^2 and |9-5| = 2^2.
a(428) = 1 since 428 = 9^2 + 13^2 + 3^2 + 13^2 with 9 = 3^2 and |9-13| = 2^2.
a(568) = 1 since 568 = 4^2 + 8^2 + 2^2 + 22^2 with 4 = 2^2 and |4-8| = 2^2.
a(632) = 1 since 632 = 16^2 + 12^2 + 6^2 + 14^2 with 16 = 4^2 and |16-12| = 2^2.
a(1144) = 1 since 1144 = 16^2 + 20^2 + 2^2 + 22^2 with 16 = 4^2 and |16-20| = 2^2.
a(1544) = 1 since 1544 = 0^2 + 0^2 + 10^2 + 38^2 with 0 = 0^2 and |0-0| = 0^2.

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.

Cf. A000118, A000290, A270969, A271775, A281939, A281941, A281976, A281977.

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

A273915 Number of ordered ways to write n as w^5 + x^2 + y^2 + z^2, where w,x,y,z are nonnegative integers with x <= y <= z.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Jun 03 2016

nonn

Let c be 1 or 4. Then any nonnegative integer n can be written as c*w^5 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers. We now prove this by induction on n. For n < 2^(10) this can be verified directly via a computer. If 2^(10) divides n, then by the induction hypothesis we can write n/2^(10) as c*w^5 + x^2 + y^2 + z^2 with w,x,y,z, nonnegative integers, and hence n = c*(2^2*w)^5 + (2^5*x)^2 + (2^5*y)^2 + (2^5*z)^2. If n is not of the form 4^k*(8m+7) with k and m nonnegative integers, then n is the sum of three squares and hence n = c*0^5 + x^2 + y^2 + z^2 for some integers x,y,z. When n = 4^k*(8m+7) > 2^(10) with k < 5, it is easy to see that n - c*1^5 or n - c*2^5 is the sum of three squares.
For any positive integer k and for each c = 2, 6, any natural number n can be written as c*w^k + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers. In fact, for every n = 0,1,2,... either n - c*0^k or n - c*1^k can be written as the sum of three squares.
See also  A270969 and  A273429 for similar results.
For some conjectural refinements of Lagrange's four-square theorem, one may consult the author's preprint arXiv:1604.06723

			a(0) = 1 since 0 = 0^5 + 0^2 + 0^2 + 0^2.
a(7) = 1 since 7 = 1^5 + 1^2 + 1^2 + 2^2.
a(8) = 1 since 8 = 0^5 + 0^2 + 2^2 + 2^2.
a(15) = 1 since 15 = 1^5 + 1^2 + 2^2 + 3^2.
a(16) = 1 since 16 = 0^5 + 0^2 + 0^2 + 4^2.
a(23) = 1 since 23 = 1^5 + 2^2 + 3^2 + 3^2.
a(24) = 1 since 24 = 0^2 + 2^2 + 2^2 + 4^2.
a(31) = 1 since 31 = 1^5 + 1^2 + 2^2 + 5^2.
a(47) = 1 since 47 = 1^5 + 1^2 + 3^2 + 6^2.
a(71) = 1 since 71 = 1^5 + 3^2 + 5^2 + 6^2.
a(79) = 1 since 79 = 1^5 + 2^2 + 5^2 + 7^2.
a(92) = 1 since 92 = 1^5 + 1^2 + 3^2 + 9^2.
a(112) = 1 since 112 = 2^5 + 0^2 + 4^2 + 8^2.
a(143) = 1 since 143 = 1^5 + 5^2 + 6^2 + 9^2.
a(191) = 1 since 191 = 1^5 + 3^2 + 9^2 + 10^2.
a(240) = 1 since 240 = 2^5 + 0^2 + 8^2 + 12^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.NT], 2016-2017.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Cf. A000118, A000290, A000584, A270969, A273429, A273917.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-w^5-x^2-y^2],r=r+1],{w,0,n^(1/5)},{x,0,Sqrt[(n-w^5)/3]},{y,x,Sqrt[(n-w^5-x^2)/2]}];Print[n," ",r];Label[aa];Continue,{n,0,80}]

A282494 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y(y+240z) a positive square, where x,y,z,w are nonnegative integers.

Original entry on oeis.org

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

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-2*z) = 0. Whether z = 0 or y = 2*z, the number y*(y+240*z) is definitely a square.
See also A282463 and A282495 for similar conjectures.

			a(3) = 1 since 3 = 1^4 + 1^2 + 0^2 + 1^2 with 1*(1+240*0) = 1^2.
a(4) = 1 since 4 = 0^4 + 2^2 + 0^2 + 0^2 with 2*(2+240*0) = 2^2.
a(39) = 1 since 39 = 1^4 + 2^2 + 3^2 + 5^2 with 2*(2+240*3) = 38^2.
a(188) = 1 since 188 = 3^4 + 5^2 + 1^2 + 9^2 with 5*(5+240*1) = 35^2.
a(399) = 1 since 399 = 3^4 + 10^2 + 7^2 + 13^2 with 10*(10+240*7) = 130^2.
a(428) = 1 since 428 = 0^4 + 10^2 + 2^2 + 18^2 with 10*(10+240*2) = 70^2.
a(439) = 1 since 439 = 1^4 + 10^2 + 7^2 + 17^2 with 10*(10+240*7) = 130^2.
a(508) = 1 since 508 = 1^4 + 5^2 + 11^2 + 19^2 with 5*(5+240*11) = 115^2.
a(748) = 1 since 748 = 3^4 + 1^2 + 21^2 + 15^2 with 1*(1+240*21) = 71^2.
a(1468) = 1 since 1468 = 2^4 + 10^2 + 26^2 + 26^2 with 10*(10+240*26) = 250^2.
a(2828) = 1 since 2828 = 3^4 + 5^2 + 11^2 + 51^2 with 5*(5+240*11) = 115^2.

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, A282463, A282495.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[y*(y+240*z)],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}]

A282542 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 + 3y + 5z and (at least) one of y,z,w are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 17 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,....
This is stronger than the 1-3-5 conjecture (cf. A271518).
By the linked JNT paper, any nonnegative integer can be expressed as the sum of a fourth power and three squares.

			a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 + 3*1 + 5*1 = 3^2 and 1 = 1^2.
a(28) = 1 since 28 = 1^2 + 1^2 + 1^2 + 5^2 with 1 + 3*1 + 5*1 = 3^2 and 1 = 1^2.
a(47) = 1 since 47 = 3^2 + 1^2 + 6^2 + 1^2 with 3 + 3*1 + 5*6 = 6^2 and 1 = 1^2.
a(92) = 1 since 92 = 1^2 + 1^2 + 9^2 + 3^2 with 1 + 3*1 + 5*1 = 3^2 and 9 = 3^2.
a(188) = 1 since 188 = 7^2 + 9^2 + 3^2 + 7^2 with 7 + 3*9 + 5*3 = 7^2 and 9 = 3^2.
a(248) = 1 since 248 = 10^2 + 2^2 + 0^2 + 12^2 with 10 + 3*2 + 5*0 = 4^2 and 0 = 0^2.
a(388) = 1 since 388 = 13^2 + 1^2 + 13^2 + 7^2 with 13 + 3*1 + 5*13 = 9^2 and 1 = 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.

Cf. A000118, A000290, A270969, A271518, A281976.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&(SQ[y]||SQ[z]||SQ[Sqrt[n-x^2-y^2-z^2]])&&SQ[x+3y+5z],r=r+1],{x,0,n^(1/2)},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,0,80}]

A272979 Number of ways to write n as x^2 + 2y^2 + 3z^3 + 4*w^4 with x,y,z,w nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Jul 13 2016

nonn

Conjecture: For positive integers a,b,c,d, any natural number can be written as a*x^2 + b*y^2 + c*z^3 + d*w^4 with x,y,z,w nonnegative integers, if and only if (a,b,c,d) is among the following 49 quadruples: (1,2,1,1), (1,3,1,1), (1,6,1,1), (2,3,1,1), (2,4,1,1), (1,1,2,1), (1,4,2,1), (1,2,3,1), (1,2,4,1), (1,2,12,1), (1,1,1,2), (1,2,1,2), (1,3,1,2), (1,4,1,2), (1,5,1,2), (1,11,1,2), (1,12,1,2), (2,4,1,2), (3,5,1,2), (1,1,4,2), (1,1,1,3), (1,2,1,3), (1,3,1,3), (1,2,4,3), (1,2,1,4), (1,3,1,4), (2,3,1,4), (1,1,2,4), (1,2,2,4), (1,8,2,4), (1,2,3,4), (1,1,1,5), (1,2,1,5), (2,3,1,5), (2,4,1,5), (1,3,2,5), (1,1,1,6), (1,3,1,6), (1,1,2,6), (1,2,1,8), (1,2,4,8), (1,2,1,10), (1,1,2,10), (1,2,1,11), (2,4,1,11), (1,2,1,12), (1,1,2,13), (1,2,1,14),(1,2,1,15).

See also A262824, A262827, A262857 and A273917 for similar conjectures.

			a(0) = 1 since 0 = 0^2 + 2*0^2 + 3*0^3 + 4*0^4.
a(1) = 1 since 1 = 1^2 + 2*0^2 + 3*0^3 + 4*0^4.
a(2) = 1 since 2 = 0^2 + 2*1^2 + 3*0^3 + 4*0^4.
a(14) = 1 since 14 = 3^2 + 2*1^2 + 3*1^3 + 4*0^4.
a(17) = 1 since 17 = 3^2 + 2*2^2 + 3*0^3 + 4*0^4.
a(59) = 1 since 59 = 3^2 + 2*5^2 + 3*0^3 + 4*0^4.
a(63) = 1 since 63 = 3^2 + 2*5^2 + 3*0^2 + 4*1^4.
a(287) = 1 since 287 = 11^2 + 2*9^2 + 3*0^2 + 4*1^4.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000290, A000578, A000583, A262824, A262827, A262857, A270969, A273429, A273915, A273917.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-4w^4-3z^3-2y^2],r=r+1],{w,0,(n/4)^(1/4)},{z,0,((n-4w^4)/3)^(1/3)},{y,0,((n-4w^4-3z^3)/2)^(1/2)}];Print[n," ",r];Continue,{n,0,100}]

A273616 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (3x^2+13y^2)*z a square, where x,y,z,w are nonnegative integers.

Original entry on oeis.org

1, 4, 4, 2, 5, 8, 4, 2, 4, 8, 11, 4, 2, 10, 8, 1, 4, 12, 10, 8, 9, 8, 9, 1, 4, 17, 16, 6, 3, 16, 8, 1, 4, 8, 18, 10, 8, 12, 13, 2, 10, 18, 9, 8, 5, 17, 11, 3, 2, 15, 22, 7, 13, 15, 17, 4, 6, 10, 11, 14, 2, 18, 17, 1, 5, 23, 13, 9, 13, 14, 14, 1, 8, 16, 26, 8, 4, 16, 7, 1, 8

Offset: 0

Zhi-Wei Sun, May 26 2016

nonn

Conjecture: For each ordered pair (a,b) = (3,13), (5,11), (15,57), (15,165), (138,150), any natural number can be written as x^2 + y^2 + z^2 + w^2 with (a*x^2+b*y^2)*z a square, where x,y,z,w are nonnegative integers.

For more conjectural refinements of Lagrange's four-square theorem, see the author's preprint arXiv:1604.06723.

			a(15) = 1 since 15 = 2^2 + 1^2 + 1^2 + 3^2 with (3*2^2+13*1^2)*1 = 5^2.
a(23) = 1 since 23 = 3^2 + 3^2 + 1^2 + 2^2 with (3*3^2+13*3^2)*1 = 12^2.
a(31) = 1 since 31 = 2^2 + 1^2 + 1^2 + 5^2 with (3*2^2+13*1^2)*1 = 5^2.
a(63) = 1 since 63 = 6^2 + 1^2 + 1^2 + 5^2 with (3*6^2+13*1^2)*1 = 11^2.
a(71) = 1 since 71 = 6^2 + 3^2 + 1^2 + 5^2 with (3*6^2+13*3^2)*1 = 15^2.
a(79) = 1 since 79 = 5^2 + 3^2 + 3^2 + 6^2 with (3*5^2+13*3^2)*3 = 24^2.
a(223) = 1 since 223 = 2^2 + 13^2 + 1^2 + 7^2 with (3*2^2+13*13^2)*1 = 47^2.
a(303) = 1 since 303 = 2^2 + 13^2 + 9^2 + 7^2 with (3*2^2+13*13^2)*9 = 141^2.
a(2703) = 1 since 2703 = 15^2 + 25^2 + 22^2 + 37^2 with (3*15^2+13*25^2)*22 = 440^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273429, A273432, A273458, A273568.

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

A273917 Number of ordered ways to write n as w^2 + 3*x^2 + y^4 + z^5, where w is a positive integer and x,y,z are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jun 04 2016

nonn

Conjectures:
(i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 7, 11, 12, 15, 19, 24, 27, 31, 34, 35, 43, 46, 47, 56, 70, 71, 72, 87, 88, 115, 136, 137, 147, 167, 168, 178, 207, 235, 236, 267, 286, 297, 423, 537, 747, 762, 1017.
(ii) Any positive integer n can be written as w^2 + x^4 + y^5 + pen(z), where w is a positive integer, x,y,z are nonnegative integers, and pen(z) denotes the pentagonal number z*(3*z-1)/2.
Conjectures a(n) > 0 and (ii) verified up to 10^11. - Mauro Fiorentini, Jul 19 2023
See also A262813, A262857, A270566, A271106 and A271325 for some other conjectures on representations.

			a(1) = 1 since 1 = 1^2 + 3*0^2 + 0^4 + 0^5.
a(3) = 1 since 3 = 1^2 + 3*0^2 + 1^4 + 1^5.
a(7) = 1 since 7 = 2^2 + 3*1^2 + 0^4 + 0^5.
a(11) = 1 since 11 = 3^2 + 3*0^2 + 1^4 + 1^5.
a(12) = 1 since 12 = 3^2 + 3*1^2 + 0^4 + 0^5.
a(15) = 1 since 15 = 1^2 + 3*2^2 + 1^4 + 1^5.
a(19) = 1 since 19 = 4^2 + 3*1^2 + 0^4 + 0^5.
a(24) = 1 since 24 = 2^2 + 3*1^2 + 2^4 + 1^5.
a(27) = 1 since 27 = 5^2 + 3*0^2 + 1^4 + 1^5.
a(31) = 1 since 31 = 2^2 + 3*3^2 + 0^4 + 0^5.
a(34) = 1 since 34 = 1^2 + 3*0^2 + 1^4 + 2^5.
a(35) = 1 since 35 = 4^2 + 3*1^2 + 2^4 + 0^5.
a(43) = 1 since 43 = 4^2 + 3*3^2 + 0^4 + 0^5.
a(46) = 1 since 46 = 1^2 + 3*2^2 + 1^4 + 2^5.
a(47) = 1 since 47 = 2^2 + 3*3^2 + 2^4 + 0^5.
a(56) = 1 since 56 = 6^2 + 3*1^2 + 2^4 + 1^5.
a(70) = 1 since 70 = 5^2 + 3*2^2 + 1^4 + 2^5.
a(71) = 1 since 71 = 6^2 + 3*1^2 + 0^4 + 2^5.
a(72) = 1 since 72 = 6^2 + 3*1^2 + 1^4 + 2^5.
a(87) = 1 since 87 = 6^2 + 2*1^2 + 2^4 + 2^5.
a(88) = 1 since 88 = 2^2 + 3*1^2 + 3^4 + 0^5.
a(115) = 1 since 115 = 8^2 + 3*1^2 + 2^4 + 2^5.
a(136) = 1 since 136 = 10^2 + 3*1^2 + 1^4 + 2^5.
a(137) = 1 since 137 = 11^2 + 3*0^2 + 2^4 + 0^5.
a(147) = 1 since 147 = 12^2 + 3*1^2 + 0^4 + 0^5.
a(167) = 1 since 167 = 2^2 + 3*7^2 + 2^4 + 0^5.
a(168) = 1 since 168 = 2^2 + 3*7^2 + 2^4 + 1^5.
a(178) = 1 since 178 = 7^2 + 3*4^2 + 3^4 + 0^5.
a(207) = 1 since 207 = 10^2 + 3*5^2 + 0^4 + 2^5.
a(235) = 1 since 235 = 12^2 + 3*5^2 + 2^4 + 0^5.
a(236) = 1 since 236 = 12^2 + 3*5^2 + 2^4 + 1^5.
a(267) = 1 since 267 = 12^2 + 3*5^2 + 2^4 + 2^5.
a(286) = 1 since 286 = 4^2 + 3*3^2 + 0^4 + 3^5.
a(297) = 1 since 297 = 3^2 + 3*0^2 + 4^4 + 2^5.
a(423) = 1 since 423 = 11^2 + 3*10^2 + 1^4 + 1^5.
a(537) = 1 since 537 = 21^2 + 3*4^2 + 2^4 + 2^5.
a(747) = 1 since 747 = 11^2 + 3*0^2 + 5^4 + 1^5.
a(762) = 1 since 762 = 27^2 + 3*0^2 + 1^4 + 2^5.
a(1017) = 1 since 1017 = 27^2 + 3*0^2 + 4^4 + 2^5.

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.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. (See Remark 1.1.)
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. (See Conjecture 3.2.)

Cf. A000118, A000290, A000326, A000583, A000584, A262813, A262827, A262857, A270566, A270969, A271076, A271106, A273429, A273915.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-3*x^2-y^4-z^5],r=r+1],{x,0,Sqrt[(n-1)/3]},{y,0,(n-1-3x^2)^(1/4)},{z,0,(n-1-3x^2-y^4)^(1/5)}];Print[n," ",r];Continue,{n,1,80}]

cp's OEIS Frontend

A282463 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers, x == y (mod 2) and z <= w such that both x and x^2 + 62*x*y + y^2 are squares.

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A273458 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x-y+z+w a nonnegative cube, where x,y,z,w are integers with x >= y >= 0 and x >= |z| <= |w|.

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A273568 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w + x + 2*y - 4*z twice a nonnegative cube, where w is an integer and x,y,z are nonnegative integers.

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A281975 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 both x and |x-y| are squares.

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A273915 Number of ordered ways to write n as w^5 + x^2 + y^2 + z^2, where w,x,y,z are nonnegative integers with x <= y <= z.

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A282494 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y*(y+240*z) a positive square, where x,y,z,w are nonnegative integers.

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A282542 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 + 3*y + 5*z and (at least) one of y,z,w are squares.

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A272979 Number of ways to write n as x^2 + 2*y^2 + 3*z^3 + 4*w^4 with x,y,z,w nonnegative integers.

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A282463 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers, x == y (mod 2) and z <= w such that both x and x^2 + 62xy + y^2 are squares.

A273568 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w + x + 2y - 4z twice a nonnegative cube, where w is an integer and x,y,z are nonnegative integers.

A282494 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y(y+240z) a positive square, where x,y,z,w are nonnegative integers.

A282542 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 + 3y + 5z and (at least) one of y,z,w are squares.

A272979 Number of ways to write n as x^2 + 2y^2 + 3z^3 + 4*w^4 with x,y,z,w nonnegative integers.

A273616 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (3x^2+13y^2)*z a square, where x,y,z,w are nonnegative integers.