A275301 -id:A275301 - cp's OEIS frontend

A275344 Number of ordered ways to write n as x^2 + y^2 + z^2 + 2w^2 with x + 2y + 3*z a square, where x,y,z,w are nonnegative integers.

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

Offset: 0

Zhi-Wei Sun, Jul 24 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 3, 5, 7, 14, 15, 16, 25, 30, 33, 84, 169, 225.
(ii) For each ordered pair (a,b) = (1,2), (1,3), (1,12), (1,23), (2,3), (2,4), (2,6), (2,7), (2,15), (2,16), any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers such that a*x + b*y is a square.
This is similar to the conjecture in A271518. It is known that any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers.

			a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 2*0^2 with 0 + 2*0 + 3*0 = 0^2.
a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 2*0^2 with 1 + 2*0 + 3*0 = 1^2.
a(3) = 1 since 3 = 1^2 + 0^2 + 0^2 + 2*1^2 with 1 + 2*0 + 3*0 = 1^2.
a(5) = 1 since 5 = 2^2 + 1^2 + 0^2 + 2*0^2 with 2 + 2*1 + 3*0 = 2^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 0^2 + 2*1^2 with 2 + 2*1 + 3*0 = 2^2.
a(14) = 1 since 14 = 1^2 + 1^2 + 2^2 + 2*2^2 with 1 + 2*1 + 3*2 = 3^2.
a(15) = 1 since 15 = 3^2 + 0^2 + 2^2 + 2*1^2 with 3 + 2*0 + 3*2 = 3^2.
a(16) = 1 since 16 = 4^2 + 0^2 + 0^2 + 2*0^2 with 4 + 2*0 + 3*0 = 2^2.
a(25) = 1 since 25 = 1^2 + 4^2 + 0^2 + 2*2^2 with 1 + 2*4 + 3*0 = 3^2.
a(30) = 1 since 30 = 3^2 + 2^2 + 3^2 + 2*2^2 with 3 + 2*2 + 3*3 = 4^2.
a(33) = 1 since 33 = 1^2 + 0^2 + 0^2 + 2*4^2 with 1 + 2*0 + 3*0 = 1^2.
a(84) = 1 since 84 = 4^2 + 6^2 + 0^2 + 2*4^2 with 4 + 2*6 + 3*0 = 4^2.
a(169) = 1 since 169 = 10^2 + 6^2 + 1^2 + 2*4^2 with 10 + 2*6 + 3*1 = 5^2.
a(225) = 1 since 225 = 10^2 + 6^2 + 9^2 + 2*2^2 with 10 + 2*6 + 3*9 = 7^2.

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. A000290, A271518, A275297, A275301.

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

A275409 Number of ordered ways to write n as 2w^2 + x^2 + y^2 + z^2 with w + x + 2y + 4*z a square, where w,x,y,z are nonnegative integers.

Original entry on oeis.org

1, 2, 1, 0, 2, 2, 2, 1, 1, 1, 0, 3, 1, 2, 1, 1, 3, 2, 5, 3, 4, 3, 1, 1, 1, 1, 2, 2, 2, 4, 2, 2, 4, 2, 7, 3, 1, 6, 2, 1, 2, 3, 4, 5, 1, 1, 3, 5, 3, 3, 4, 3, 7, 3, 2, 4, 3, 4, 4, 3, 1, 4, 5, 3, 6, 4, 4, 4, 5, 7, 7, 3, 6, 5, 5, 4, 3, 11, 2, 2, 4

Offset: 0

Zhi-Wei Sun, Jul 26 2016

nonn

Conjecture: (i) a(n) > 0 except for n = 3, 10, and a(n) = 1 only for n = 0, 2, 7, 8, 9, 12, 14, 15, 22, 23, 24, 25, 36, 39, 44, 45, 60, 87, 98, 106, 110, 111, 183.
(ii) Any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers such that x + 2*y + 3*z - 3*w is a square.
(iii) For each triple (a,b,c) = (1,2,1), (1,2,3), (1,3,1), (2,4,1), (2,4,2), (2,4,3), (2,4,4), (2,4,8), (8,9,5), any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers such that a*x + b*y - c*z is a square.
(iv) Any natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w nonnegative integers such that x + 2*y - 2*z is twice a nonnegative cube. Also, each natural number can be written as x^2 + y^2 + z^2 + 2*w^3 with x,y,z,w nonnegative integers such that x + 3*y - z is a square.
See also A275344 and A275301 for related conjectures. We are able to show that each natural number can be written as x^2 + y^2 + z^2 + 2*w^2 with x,y,z,w integers such that x + y + z = t^2 for some t = 0, 1, 2.

			a(2) = 1 since 2 = 2*1^2 + 0^2 + 0^2 + 0^2 with 1 + 0 + 2*0 + 4*0 = 1^2.
a(7) = 1 since 7 = 2*1^2 + 0^2 + 2^2 + 1^2 with 1 + 0 + 2*2 + 4*1 = 3^2.
a(8) = 1 since 8 = 2*1^2 + 2^2 + 1^2 + 1^2 with 1 + 2 + 2*1 + 4*1 = 3^2.
a(9) = 1 since 9 = 2*2^2 + 0^2 + 1^2 + 0^2 with 2 + 0 + 2*1 + 4*0 = 2^2.
a(12) = 1 since 12 = 2*2^2 + 2^2 + 0^2 + 0^2 with 2 + 2 + 2*0 + 4*0 = 2^2.
a(14) = 1 since 14 = 2*0^2 + 2^2 + 1^2 + 3^2 with 0 + 2 + 2*1 + 4*3 = 4^2.
a(15) = 1 since 15 = 2*1^2 + 2^2 + 3^2 + 0^2 with 1 + 2 + 2*3 + 4*0 = 3^2.
a(22) = 1 since 22 = 2*1^2 + 4^2 + 2^2 + 0^2 with 1 + 4 + 2*2 + 4*0 = 3^2.
a(23) = 1 since 23 = 2*3^2 + 2^2 + 0^2 + 1^2 with 3 + 2 + 2*0 + 4*1 = 3^2.
a(24) = 1 since 24 = 2*0^2 + 4^2 + 2^2 + 2^2 with 0 + 4 + 2*2 + 4*2 = 4^2.
a(25) = 1 since 25 = 2*0^2 + 4^2 + 0^2 + 3^2 with 0 + 4 + 2*0 + 4*3 = 4^2.
a(36) = 1 since 36 = 2*3^2 + 1^2 + 4^2 + 1^2 with 3 + 1 + 2*4 + 4*1 = 4^2.
a(39) = 1 since 39 = 2*1^2 + 6^2 + 1^2 + 0^2 with 1 + 6 + 2*1 + 4*0 = 3^2.
a(44) = 1 since 44 = 2*3^2 + 0^2 + 1^2 + 5^2 with 3 + 0 + 2*1 + 4*5 = 5^2.
a(45) = 1 since 45 = 2*0^2 + 5^2 + 2^2 + 4^2 with 0 + 5 + 2*2 + 4*4 = 5^2.
a(60) = 1 since 60 = 2*2^2 + 6^2 + 4^2 + 0^2 with 2 + 6 + 2*4 + 4*0 = 4^2.
a(87) = 1 since 87 = 2*3^2 + 2^2 + 8^2 + 1^2 with 3 + 2 + 2*8 + 4*1 = 5^2.
a(98) = 1 since 98 = 2*4^2 + 1^2 + 8^2 + 1^2 with 4 + 1 + 2*8 + 4*1 = 5^2.
a(106) = 1 since 106 = 2*2^2 + 8^2 + 3^2 + 5^2 with 2 + 8 + 2*3 + 4*5 = 6^2.
a(110) = 1 since 110 = 2*6^2 + 5^2 + 3^2 + 2^2 with 6 + 5 + 2*3 + 4*2 = 5^2.
a(111) = 1 since 111 = 2*5^2 + 3^2 + 6^2 + 4^2 with 5 + 3 + 2*6 + 4*4 = 6^2.
a(183) = 1 since 183 = 2*3^2 + 10^2 + 4^2 + 7^2 with 3 + 10 + 2*4 + 4*7 = 7^2.

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. A000290, A271518, A275297, A275301, A275344.

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

A338263 Number of ways to write 8n+7 as 2w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w(3x+7y+10z) is a square and also one of w, x, y, z is a square.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Oct 27 2020

nonn

Conjecture 1: Each nonnegative integer can be written as 2*w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w*(3*x+7*y+10*z) is a square and one of w, x, y, z is also a square.
As all the nonnegative integers not of the form 4^k*(8*m+7) (k>=0, m>=0) can be written as 2*0^2 + x^2 + y^2 + z^2 with x, y, z integers, Conjecture 1 has the following equivalent version: a(n) > 0 for all n = 0,1,...
We have verified that a(n) > 0 for all n = 0..10^5.
Conjecture 2: If (a,b) is among the ordered pairs (1,2), (1,3), (2,4), (2,5), (2,8), (2,24), (6,8), (6,32), (9,12) and (18,24), then each n = 0,1,... can be written as 2*w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w*(a*x+b*y) is a square.

			a(7) = 1, and 8*7+7 = 63 = 2*3^2 + 6^2 + 0^2 + 3^2 with 0 = 0^2 and 3*(3*6+7*0+10*3) = 12^2.
a(11) = 1, and 8*11+7 = 95 = 2*1^2 + 2^2 + 5^2 + 8^2 with 1 = 1^2 and 1*(3*2+7*5+10*8) = 11^2.
a(15) = 1, and 8*15+7 = 127 = 2*7^2 + 3^2 + 2^2 + 4^2 with 4 = 2^2 and 7*(3*3+7*2+10*4) = 21^2.
a(64) = 1, and 8*64+7 = 519 = 2*3^2 + 1^2 + 20^2 + 10^2 with 1 = 1^2 and 3*(3*1+7*20+10*10) = 27^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..6000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. See also arXiv:1604.06723 [math.NT].
Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893. See also arXiv:1701.05868 [math.NT].
Zhi-Wei Sun, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.

Cf. A000290, A271518, A271724, A275301, A275344.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[8n+7-2w^2-x^2-y^2]&&(SQ[w]||SQ[x]||SQ[y]||SQ[8n+7-2w^2-x^2-y^2])&&SQ[w(3x+7y+10*Sqrt[8n+7-2w^2-x^2-y^2])],r=r+1],{w,0,Sqrt[4n+3]},{x,0,Sqrt[8n+7-2w^2]},{y,0,Sqrt[8n+7-2w^2-x^2]}];tab=Append[tab,r],{n,0,80}];tab

cp's OEIS Frontend

A275344 Number of ordered ways to write n as x^2 + y^2 + z^2 + 2w^2 with x + 2y + 3*z a square, where x,y,z,w are nonnegative integers.

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A275409 Number of ordered ways to write n as 2w^2 + x^2 + y^2 + z^2 with w + x + 2y + 4*z a square, where w,x,y,z are nonnegative integers.

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A338263 Number of ways to write 8n+7 as 2w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w(3x+7y+10z) is a square and also one of w, x, y, z is a square.

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cp's OEIS Frontend

A275344 Number of ordered ways to write n as x^2 + y^2 + z^2 + 2*w^2 with x + 2*y + 3*z a square, where x,y,z,w are nonnegative integers.

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A275409 Number of ordered ways to write n as 2*w^2 + x^2 + y^2 + z^2 with w + x + 2*y + 4*z a square, where w,x,y,z are nonnegative integers.

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A338263 Number of ways to write 8*n+7 as 2*w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w*(3*x+7*y+10*z) is a square and also one of w, x, y, z is a square.

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A275344 Number of ordered ways to write n as x^2 + y^2 + z^2 + 2w^2 with x + 2y + 3*z a square, where x,y,z,w are nonnegative integers.

A275409 Number of ordered ways to write n as 2w^2 + x^2 + y^2 + z^2 with w + x + 2y + 4*z a square, where w,x,y,z are nonnegative integers.

A338263 Number of ways to write 8n+7 as 2w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w(3x+7y+10z) is a square and also one of w, x, y, z is a square.