A275299 -id:A275299 - cp's OEIS frontend

A275297 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^3 with x + 2*y a square, where x,y,z,w are nonnegative integers with z >= w.

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

Offset: 0

Zhi-Wei Sun, Jul 22 2016

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 3, 7, 8, 11, 12, 15, 23, 24, 32, 39, 47, 71, 103, 120, 136, 159, 176, 183, 218, 359, 463.
Compare this conjecture with Conjecture 5.1 of the author's preprint arXiv:1604.06723. See also A275298 and A275299 for similar conjectures.
By Theorem 1.1 of arXiv:1604.06723, any natural number can be written as the sum of three squares and a sixth power.
Let c be 1 or 2. By the conjecture in A272979, any n = 0,1,2,... can be written as x^2 + 2*y^2 + z^3 + 2*c^2*w^4 with x,y,z,w nonnegative integers, and hence n = x^2 + (y+c*w^2)^2 + (y-c*w^2)^2 + z^3 with (y+c*w^2)-(y-c*w^2) = 2*c*w^2. If n > 0 is not among the 174 terms in the b-file of A275169, then the conjecture in A275169 implies that n can be written as x^2 + y^2 + z^2 + w^3 with x - y = 0^2, where x,y,z,w are nonnegative integers. If n is among the 174 terms in the b-file of A275169, then we may use a computer to verify that n can be written as x^2 + y^2 + z^2 + w^3 with c*(x-y) a square, where x,y,z,w are nonnegative integers.

			a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 0^3 with 0 + 2*0 = 0^2 and 0 = 0.
a(1) = 2 since 1 = 0^2 + 0^2 + 1^2 + 0^3 with 0 + 2*0 = 0^2 and 1 > 0, and also 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 + 2*0 = 1^2 and 0 = 0.
a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^3 with 1 + 2*0 = 1^2 and 1 = 1.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^3 with 2 + 2*1 = 2^2 and 1 = 1.
a(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^3 with 0 + 2*2 = 2^2 and 2 > 0.
a(11) = 1 since 11 = 1^2 + 0^2 + 3^2 + 1^3 with 1 + 2*0 = 1^2 and 3 > 1.
a(12) = 1 since 12 = 0^2 + 0^2 + 2^2 + 2^3 with 0 + 2*0 = 0^2 and 2 = 2.
a(15) = 1 since 15 = 2^2 + 1^2 + 3^2 + 1^3 with 2 + 2*1 = 2^2 and 3 > 1.
a(23) = 1 since 23 = 3^2 + 3^2 + 2^2 + 1^3 with 3 + 2*3 = 3^2 and 2 > 1.
a(24) = 1 since 24 = 0^2 + 0^2 + 4^2 + 2^3 with 0 + 2*0 = 0^2 and 4 > 2.
a(32) = 1 since 32 = 4^2 + 0^2 + 4^2 + 0^3 with 4 + 2*0 = 2^2 and 4 > 0.
a(39) = 1 since 39 = 5^2 + 2^2 + 3^2 + 1^3 with 5 + 2*2 = 3^2 and 3 > 1.
a(47) = 1 since 47 = 0^2 + 2^2 + 4^2 + 3^3 with 0 + 2*2 = 2^2 and 4 > 3.
a(71) = 1 since 71 = 6^2 + 5^2 + 3^2 + 1^3 with 6 + 2*5 = 4^2 and 3 > 1.
a(103) = 1 since 103 = 2^2 + 7^2 + 7^2 + 1^3 with 2 + 2*7 = 4^2 and 7 > 1.
a(120) = 1 since 120 = 5^2 + 2^2 + 8^2 + 3^3 with 5 + 2*2 = 3^2 and 8 > 3.
a(136) = 1 since 136 = 0^2 + 8^2 + 8^2 + 2^3 with 0 + 2*8 = 4^2 and 8 > 2.
a(159) = 1 since 159 = 10^2 + 3^2 + 7^2 + 1^3 with 10 + 2*3 = 4^2 and 7 > 1.
a(176) = 1 since 176 = 2^2 + 1^2 + 12^2 + 3^3 with 2 + 2*1 = 2^2 and 12 > 3.
a(183) = 1 since 183 = 6^2 + 5^2 + 11^2 + 1^3 with 6 + 2*5 = 4^2 and 11 > 1.
a(218) = 1 since 218 = 5^2 + 2^2 + 8^2 + 5^3 with 5 + 2*2 = 3^2 and 8 > 5.
a(359) = 1 since 359 = 11^2 + 7^2 + 8^2 + 5^3 with 11 + 2*7 = 5^2 and 8 > 5.
a(463) = 1 since 463 = 2^2 + 17^2 + 13^2 + 1^3 with 2 + 2*17 = 6^2 and 13 > 1.

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, A000578, A271518, A272979, A273404, A273429, A275169, A275298, A275299.

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

A275298 Number of ordered ways to write n as w^3 + x^2 + y^2 + z^2 with x - w a square, where x,y,z,w are nonnegative integers with y <= z > w.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jul 22 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 4, 7, 8, 12, 16, 23, 24, 40, 47, 71, 167, 311, 599.
(ii) For each triple (a,b,c) = (1,1,1), (2,1,1), (2,1,2), (2,2,2), (3,1,2), any natural number can be written as x^2 + y^2 + z^2 + w^3 with x,y,z,w nonnegative integers such that a*y - b*z - c*w is a square.
See also A275297 and A275299 for similar conjectures.

			a(1) = 1 since 1 = 0^3 + 0^2 + 0^2 + 1^2 with 0 - 0 = 0^2 and 0 < 1 > 0.
a(3) = 1 since 3 = 0^3 + 1^2 + 1^2 + 1^2 with 1 - 0 = 1^2 and 1 = 1 > 0.
a(4) = 1 since 4 = 0^3 + 0^2 + 0^2 + 2^2 with 0 - 0 = 0^2 and 0 < 2 > 0.
a(7) = 1 since 7 = 1^3 + 1^2 + 1^2 + 2^2 with 1 - 1 = 0^2 and 1 < 2 > 1.
a(8) = 1 since 8 = 0^3 + 0^2 + 2^2 + 2^2 with 0 - 0 = 0^2 and 2 = 2 > 0.
a(12) = 1 since 12 = 1^3 + 1^2 + 1^2 + 3^2 with 1 - 1 = 0^2 and 1 < 3 > 1.
a(16) = 1 since 16 = 0^3 + 0^2 + 0^2 + 4^2 with 0 - 0 = 0^2 and 0 < 4 > 0.
a(23) = 1 since 23 = 1^3 + 2^2 + 3^2 + 3^2 with 2 - 1 = 1^2 and 3 = 3 > 1.
a(24) = 1 since 24 = 0^3 + 4^2 + 2^2 + 2^2 with 4 - 0 = 2^2 and 2 = 2 > 0.
a(40) = 1 since 40 = 0^3 + 0^2 + 2^2 + 6^2 with 0 - 0 = 0^2 and 2 < 6 > 0.
a(47) = 1 since 47 = 1^3 + 1^2 + 3^2 + 6^2 with 1 - 1 = 0^2 and 3 < 6 > 1.
a(71) = 1 since 71 = 1^3 + 5^2 + 3^2 + 6^2 with 5 - 1 = 2^2 and 3 < 6 > 1.
a(167) = 1 since 167 = 1^3 + 2^2 + 9^2 + 9^2 with 2 - 1 = 1^2 and 9 = 9 > 1.
a(311) = 1 since 311 = 1^3 + 2^2 + 9^2 + 15^2 with 2 - 1 = 1^2 and 9 < 15 > 1.
a(599) = 1 since 599 = 5^3 + 5^2 + 7^2 + 20^2 with 5 - 5 = 0^2 and 7 < 20 > 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000290, A000578, A271518, A275297, A275299.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[r=0;Do[If[CQ[n-x^2-y^2-z^2]&&SQ[x-(n-x^2-y^2-z^2)^(1/3)]&&(n-x^2-y^2-z^2)^(1/3)

A275300 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^3 with x + y + z a square, where x,y,z are integers with x >= |y| <= |z|, and w is a nonnegative integer.

Original entry on oeis.org

1, 3, 3, 3, 2, 1, 5, 4, 3, 5, 4, 5, 1, 2, 9, 4, 4, 4, 7, 6, 1, 2, 6, 1, 7, 7, 8, 6, 3, 5, 7, 1, 7, 11, 11, 9, 4, 5, 6, 4, 3, 15, 10, 8, 2, 7, 9, 1, 4, 9, 5, 12, 5, 11, 10, 3, 8, 5, 3, 8, 7, 10, 10, 2, 4, 11, 9, 8, 6, 10, 13, 1, 7, 10, 8, 8, 2, 10, 14, 3, 10

Offset: 0

Zhi-Wei Sun, Jul 22 2016

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 5, 12, 20, 23, 31, 47, 71, 103, 148, 164.
The author proved in arXiv:1604.06723 that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x + y + z a square, where x,y,z,w are integers.
See also A275297, A275298, A275299 and A272620 for similar conjectures.

			a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 0^3 with 0 + 0 + 0 = 0^2 and 0 = 0 = 0.
a(5) = 1 since 5 = 2^2 + 0^2 + (-1)^2 + 0^3 with 2 + 0 + (-1) = 1^2 and 2 > 0 < |-1|.
a(12) = 1 since 12 = 3^2 + (-1)^2 + (-1)^2 + 1^3 with 3 + (-1) + (-1) = 1^2 and 3 > |-1| = |-1|.
a(20) = 1 since 20 = 3^2 + 1^2 + (-3)^2 + 1^3 with 3 + 1 + (-3) = 1^2 and 3 > 1 < |-3|.
a(23) = 1 since 23 = 3^2 + (-2)^2 + 3^2 + 1^3 with 3 + (-2) + 3 = 2^2 and 3 > |-2| < 3.
a(31) = 1 since 31 = 5^2 + 1^2 + (-2)^2 + 1^3 with 5 + 1 + (-2) = 2^2 and 5 > 1 < |-2|.
a(47) = 1 since 47 = 6^2 + 1^2 + (-3)^2 + 1^3 with 6 + 1 + (-3) = 2^2 and 6 > 1 < |-3|.
a(71) = 1 since 71 = 6^2 + 3^2 + (-5)^2 + 1^3 with 6 + 3 + (-5) = 2^2 and 6 > 3 < |-5|.
a(103) = 1 since 103 = 7^2 + 2^2 + 7^2 + 1^3 with 7 + 2 + 7 = 4^2 and 7 > 2 < 7.
a(148) = 1 since 148 = 9^2 + (-2)^2 + (-6)^2 + 3^3 with 9 + (-2) + (-6) = 1^2 and 9 > |-2| < |-6|.
a(164) = 1 since 164 = 9^2 + 1^2 + (-9)^2 + 1^3 with 9 + 1 + (-9) = 1^2 and 9 > 1 < |-9|.

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, A000578, A271518, A272620, A275297, A275298, A275299.

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

cp's OEIS Frontend

A275297 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^3 with x + 2*y a square, where x,y,z,w are nonnegative integers with z >= w.

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A275298 Number of ordered ways to write n as w^3 + x^2 + y^2 + z^2 with x - w a square, where x,y,z,w are nonnegative integers with y <= z > w.

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A275300 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^3 with x + y + z a square, where x,y,z are integers with x >= |y| <= |z|, and w is a nonnegative integer.

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