A271510 -id:A271510 - cp's OEIS frontend

A281659 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 9x^2 + 16y^2 + 24z^2 + 48w^2 a square, where x,y,z,w are nonnegative integers.

1, 2, 2, 2, 2, 1, 4, 1, 2, 3, 4, 1, 5, 1, 5, 1, 2, 2, 8, 1, 2, 5, 2, 2, 4, 3, 2, 8, 1, 5, 4, 2, 2, 3, 3, 4, 4, 3, 2, 2, 4, 2, 6, 1, 1, 3, 2, 3, 5, 2, 3, 7, 3, 1, 7, 1, 5, 2, 2, 4, 1, 1, 6, 7, 2, 1, 7, 2, 4, 5, 6, 0, 8, 3, 6, 8, 5, 3, 4, 2, 2

Offset: 0

Zhi-Wei Sun, Jan 26 2017

nonn

Conjecture: (i) a(n) > 0 except for n = 71, 85.
(ii) Any nonnegative integer n not among 39, 71 and 649 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 16*y^2 + 24*z^2 + 32*w^2 a square, where x,y,z,w are integers.
(iii) Each nonnegative integer n not among 5, 7, 23, 47, 93, 103, 109, 151, 191 and 911 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 3*y^2 + 8*z^2 + 16*w^2 a square, where x,y,z,w are integers.
(iv) Any nonnegative integer n not among 15, 19, 71, 103, 191 and 559 can be written as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 48*y^2 + 64*z^2 + 96*w^2 a square, where x,y,z,w are integers.
(v) Any nonnegative integer n not among 5, 93, 95, 161 and 309 can be written as x^2 + y^2 + z^2 + w^2 with 16*x^2 + 25*y^2 + 48*z^2 + 128*w^2 a square, where x,y,z,w are integers.
(vi) Each nonnegative integer not among 13, 19, 39, 41, 71, 109, 131, 193, 233, 377, 415 and 941 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 24*y^2 + 48*z^2 + 144*w^2 a square, where x,y,z,w are integers.
We have verified part (i) of the conjecture for n up to 1.2*10^5.

			a(11) = 1 since 11 = 3^2 + 1^2 + 1^2 + 0^2 with 9*3^2 + 16*1^2 + 24*1^2 + 48*0 = 11^2.
a(170) = 1 since 170 = 3^2 + 6^2 + 2^2 + 11^2 with 9*3^2 + 16*6^2 + 24*2^2 + 48*11^2 = 81^2.
a(305) = 1 since 305 = 0^2 + 15^2 + 4^2 + 8^2 with 9*0^2 + 16*15^2 + 24*4^2 + 48*8^2 = 84^2.

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, A271510, A271513, A271518.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[9x^2+16y^2+24z^2+48*(n-x^2-y^2-z^2)],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}]

A301375 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that x(y+3z) is a cube or half a cube and y <= z <= w if x = 0.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 19 2018

nonn

Conjecture: a(n) > 0 for all n > 0.

We have verified this for all n = 1..3*10^6.

			a(14) = 1 since 14 = 0^2 + 1^2 + 2^2 + 3^2 with 0*(1+3*2) = 0^3.
a(15) = 1 since 15 = 2^2 + 1^2 + 1^2 + 3^2 with 2*(1+3*1) = 2^3.
a(16) = 1 since 16 = 0^2 + 0^2 + 0^2 + 4^2 with 0*(0+3*0) = 0^3.
a(60) = 1 since 60 = 4^2 + 2^2 + 2^2 + 6^2 with 4*(2+3*2) = 4^3/2.
a(92) = 1 since 92 = 6^2 + 6^2 + 4^2 + 2^2 with 6*(6+3*4) = 6^3/2.
a(240) = 1 since 240 = 2^2 + 14^2 + 6^2 + 2^2 with 2*(14+3*6) = 4^3.
a(807) = 1 since 807 = 1^2 + 21^2 + 2^2 + 19^2 with 1*(21+3*2) = 3^3.

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, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300791, A300792, A300844, A300908, A301303, A301304, A301314.

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

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A281659 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 9x^2 + 16y^2 + 24z^2 + 48w^2 a square, where x,y,z,w are nonnegative integers.

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A301375 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that x(y+3z) is a cube or half a cube and y <= z <= w if x = 0.

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cp's OEIS Frontend

A281659 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 16*y^2 + 24*z^2 + 48*w^2 a square, where x,y,z,w are nonnegative integers.

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A301375 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that x*(y+3*z) is a cube or half a cube and y <= z <= w if x = 0.

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A281659 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 9x^2 + 16y^2 + 24z^2 + 48w^2 a square, where x,y,z,w are nonnegative integers.

A301375 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that x(y+3z) is a cube or half a cube and y <= z <= w if x = 0.