A351306 -id:A351306 - cp's OEIS frontend

A351338 Least nonnegative integer m such that n = x^3 + y^3 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

0, 0, 0, 5, 11, 4, 1, 1, 0, 0, 3, 2, 2, 35, 1, 1, 0, 7, 2, 2, 2, 12, 14, 10, 4, 1, 1, 0, 0, 3, 3, 44, 22, 1, 1, 0, 3, 3, 2, 8, 8, 127, 4, 7, 3, 2, 2, 8, 2, 2, 97, 7, 1, 1, 0, 2, 2, 2, 17, 13, 4, 4, 1, 1, 0, 0, 6, 20, 4, 4, 1, 1, 0, 15, 3, 2, 53, 22, 7, 3, 4, 6, 2, 2, 5, 14, 139, 4, 4, 1, 1, 0, 5, 3, 5, 22, 4, 3, 3, 3, 3

Offset: 0

Zhi-Wei Sun, Feb 08 2022

nonn

Conjecture: a(n) exists for any n >= 0. Equivalently, each integer can be written as x^3 + y^3 - (z^3 + w^3) with x,y,z,w nonnegative integers.

This is stronger than Sierpinski's conjecture which states that any integer is a sum of four integer cubes.

			a(41) = 127 with 41 = 41^3 + 128^3 - 49^3 -127^3.
a(130) = 143 with 130 = 37^3 + 169^3 - 125^3 - 143^3.
a(4756) = 533 with 4756 = 265^3 + 538^3 - 284^3 - 533^3.
a(5134) = 389 with 5134 = 19^3 + 418^3 - 242^3 - 389^3.

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

Cf. A000578, A004826, A004999, A351306, A351321.

CQ[n_]:=IntegerQ[n^(1/3)];
tab={};Do[m=0; Label[bb]; k=m^3; Do[If[CQ[n+k+x^3-y^3], tab=Append[tab,m];Goto[aa]],  {x, 0, m}, {y, 0, ((n+k+x^3)/2)^(1/3)}];m=m+1; Goto[bb]; Label[aa], {n, 0, 100}];Print[tab]

A351321 Least positive integer m such that m^6n = u^6 + v^3 - (x^6 + y^3) for some nonnegative integers u,v,x,y with x^6 + y^3 <= m^6n^2.

Original entry on oeis.org

1, 1, 1, 21, 6, 3, 1, 1, 1, 1, 7, 7, 3, 3, 3, 3, 2, 6, 1, 1, 1, 2, 6, 3, 5, 1, 1, 1, 1, 5, 2, 6, 12, 3, 1, 1, 1, 1, 1, 6, 6, 3, 3, 2, 1, 1, 4, 3, 2, 3, 3, 2, 7, 1, 1, 1, 1, 1, 3, 6, 1, 1, 1, 1, 1, 1, 4, 15, 3, 3, 1, 1, 1, 3, 4, 2, 3, 6, 3, 3, 2, 3, 1, 1, 3, 3, 3, 6, 1, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1

Offset: 0

Zhi-Wei Sun, Feb 07 2022

nonn

6-6-3-3 Conjecture: Each rational number can be written as u^6 - v^6 + x^3 - y^3 with u,v,x,y nonnegative rational numbers. Moreover, a(n) exists for any nonnegative integer n.

As a/b = (a*b^5)/b^6 for any integer a and nonzero integer b, the second assertion in the conjecture implies the first one.

			a(3) = 21 with 21^6*3 = 22^6 + 956^3 - (30^6 + 93^3) and 30^6 + 93^3 <= 21^6*3^2.
a(67) = 15 with 15^6*67 = 21^6 + 1091^3 - (15^6 + 848^3) and 15^6 + 848^3 <= 15^6*67^2.
a(564) = 14 with 14^6*564 = 69^6 + 4415^3 - (16^6 + 5746^3) and 16^6 + 5746^3 <= 14^6*564^2.
a(949) = 18 with 18^6*949 = 7^6 + 11784^3 - (11^6 + 11706^3) and 11^6 + 11706^3 <= 18^6*949^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..1000

Cf.A000578, A001014, A351306, A351338.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={};Do[m=1;Label[bb];k=m^6;Do[If[CQ[k*n+x^6+y^3-z^6],tab=Append[tab,m];Goto[aa]],
{x,0,m*n^(1/3)},{y,0,(k*n^2-x^6)^(1/3)},{z,0,(k*n+x^6+y^3)^(1/6)}]; m=m+1;Goto[bb]; Label[aa],{n,0,100}];Print[tab]

A351341 Least nonnegative integer m such that n = x^4 + y^4 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

Original entry on oeis.org

0, 0, 0, 63, 3, 3, 4, 2, 2, 2, 4, 21, 37, 6, 1, 1, 0, 0, 4, 11, 7, 14, 5, 2, 2, 4, 8, 3, 3, 5, 1, 1, 0, 4, 4, 45, 5, 5, 11, 6, 6, 6, 32, 3, 7, 11, 3, 3, 6, 8, 8, 48, 13, 3, 3, 3, 6, 6, 31, 20, 93, 55, 3, 49, 33, 2, 2, 5, 5, 3, 3, 4, 2, 2, 2, 69, 17, 29, 11, 1, 1, 0, 0, 5, 61, 29, 8, 5, 2, 2, 4, 21, 29, 51, 6, 1, 1, 0, 4, 85, 13

Offset: 0

Zhi-Wei Sun, Feb 08 2022

nonn

Conjecture 1: Let k be 4 or 5. Then each integer can be written as x^k + y^k - (z^3 + w^3) with x,y,z,w nonnegative integers.
Two examples for k = 5: -4 = 58^5 + 76^5 - (775^3 + 1397^3) and 14 = 40^5 + 67^5 - (125^3 + 1132^3).
Conjecture 2: Let k be among 4, 5, 6 and 7. Then any integer can be written as x^k + y^k - (z^2 + w^2) with x,y,z,w nonnegative integers.
Examples for k = 6, 7: 170 = 9^6 + 15^6 - (2114^2 + 2730^2) and 469 = 7^7 + 8^7 - (1001^2 + 1385^2).
Conjecture 3: For any integer k > 3, there are no nonnegative integers x,y,z,w such that x^k + y^k - (z^k + w^k) = 3.
See also another similar conjecture in A351338.

			a(60) = 93 with 60 = 25^4 + 27^4 - (49^3 + 93^3).
a(527) = 527 with 527 = 29^4 + 110^4 - (91^3 + 527^3).
a(2198) = 1704 with 2198 = 85^4 + 304^4 - (1539^3 + 1704^3).
a(4843) = 1965 with 4843 = 142^4 + 338^4 - (1804^3 + 1965^3).

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

Cf. A000578, A000583, A000584, A001014, A001015, A001481, A004831, A004999, A351306, A351321, A351338.

QQ[n_]:=IntegerQ[n^(1/4)];
tab={};Do[m=0;Label[bb]; k=m^3;Do[If[QQ[n+k+x^3-y^4], tab=Append[tab,m];Goto[aa]],{x,0,m},{y,0,((n+k+x^3)/2)^(1/4)}];m=m+1;Goto[bb];Label[aa],{n, 0, 100}];Print[tab]

A351312 Least positive integer m such that m^4n = x^4 + y^4 - z^2 for some nonnegative integers x,y,z with z <= m^2n.

Original entry on oeis.org

1, 1, 1, 2, 20, 2, 2, 1, 1, 24, 6, 2, 1, 1, 15, 1, 1, 1, 1, 20, 2, 2, 2, 1, 5, 2, 2, 4, 1, 17, 1, 1, 1, 1, 2, 2, 2, 2, 6, 2, 1, 1, 2, 2, 13, 1, 1, 1, 1, 1, 1, 4, 2, 2, 10, 2, 1, 1, 2, 10, 1, 1, 1, 2, 10, 1, 1, 2, 2, 20, 6, 1, 1, 1, 12, 6, 1, 1, 1, 4, 1, 1, 1, 2, 6, 2, 2, 1, 1, 5, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 2

Offset: 0

Zhi-Wei Sun, Feb 06 2022

nonn

Conjecture: Any integer m can be written as x^4 + y^4 - z^2, where x,y,z are rational numbers with z <= |m|.

This implies the existence of a(n) for all n >= 0. As a/b = (a*b^3)/b^4 for any integer a and nonzero integer b, the conjecture also implies that any rational number can be written as x^4 + y^4 - z^2 with x,y,z rational numbers.

			a(4) = 20 with 20^4*4 = 15^4 + 28^4 - 159^2 and 159 < 20^2*4.
a(9) = 24 with 24^4*9 = 20^4 + 45^4 - 1129^2 and 1129 < 24^2*9.
a(164) = 30 with 30^4*164 = 66^4 + 185^4 - 32519^2 and 32519 < 30^2*164.
From _Chai Wah Wu_, Feb 21 2022: (Start)
a(244) = 50 with 50^4*244 = 455^4 + 504^4 - 325359^2 and 325359 < 50^2*244.
a(329) = 46 with 46^4*329 = 90^4 + 195^4 - 6199^2 and 6199 < 46^2*329.
a(414) = 21 with 21^4*414 = 135^4 + 415^4 - 172954^2 and 172954 < 21^2*414.
a(554) = 74 with 74^4*554 = 475^4 + 710^4 - 537039^2 and 537039 < 74^2*554.
(End)

Chai Wah Wu, Table of n, a(n) for n = 0..2648 (terms 0..200 from Zhi-Wei Sun)
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000290, A000583, A351306.

QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
tab={};Do[m=1; Label[bb]; k=m^4; Do[If[QQ[k*n+z^2-x^4],
tab=Append[tab,m];Goto[aa]],{z,0,m^2*n},{x,0,((k*n+z^2)/2)^(1/4)}]; m=m+1; Goto[bb];Label[aa],{n,0,100}];Print[tab]

cp's OEIS Frontend

A351338 Least nonnegative integer m such that n = x^3 + y^3 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

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A351321 Least positive integer m such that m^6*n = u^6 + v^3 - (x^6 + y^3) for some nonnegative integers u,v,x,y with x^6 + y^3 <= m^6*n^2.

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A351341 Least nonnegative integer m such that n = x^4 + y^4 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

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A351312 Least positive integer m such that m^4*n = x^4 + y^4 - z^2 for some nonnegative integers x,y,z with z <= m^2*n.

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A351321 Least positive integer m such that m^6n = u^6 + v^3 - (x^6 + y^3) for some nonnegative integers u,v,x,y with x^6 + y^3 <= m^6n^2.

A351312 Least positive integer m such that m^4n = x^4 + y^4 - z^2 for some nonnegative integers x,y,z with z <= m^2n.