cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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.

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

Views

Author

Zhi-Wei Sun, Feb 07 2022

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf.A000578, A001014, A351306, A351338.

Programs

  • Mathematica
    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

Views

Author

Zhi-Wei Sun, Feb 08 2022

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    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]

A351375 Least nonnegative integer m such that n = x^5 + y^5 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

Original entry on oeis.org

0, 0, 0, 174, 3, 3, 3, 53, 53, 25, 3, 20, 93, 932, 1132, 405, 2, 2, 7, 6, 6, 972, 39712, 2, 2, 2, 6, 6, 6, 3, 1, 1, 0, 0, 271, 43, 3, 3, 14, 14, 42, 583, 131, 181, 1131, 15, 93, 176, 2, 48, 685, 6, 16, 16, 5, 2, 2, 559, 6, 6, 1351, 421, 1, 1, 0, 60, 12, 12, 300, 29, 14, 20, 53, 52
Offset: 0

Views

Author

Zhi-Wei Sun, Feb 09 2022

Keywords

Comments

Conjecture: a(n) exists for any nonnegative integer n.
See also Conjecture 1 in A351341.

Examples

			a(3) = 174 with 3 = 11^5 + 22^5 - 36^3 - 174^3.
a(13) = 932 with 13 = 12^5 + 61^5 - 328^3 - 932^3.
a(14) = 1132 with 14 = 40^5 + 67^5 - 125^3 - 1132^3.
a(15) = 405 with 15 = 31^5 + 36^5 - 283^3 - 405^3.
a(21) = 972 with 21 = 37^5 + 64^5 - 608^3 - 972^3.
From _Chai Wah Wu_, Feb 21 2022: (Start)
a(22) = 39712 with 22 = 106^5 + 599^5 - 24385^3 - 39712^3.
a(154) = 22049 with 154 = 163^5 + 432^5 - 16438^3 - 22049^3.
a(246) = 22811 with 246 = 161^5 + 412^5 - 4786^3 - 22811^3.
a(446) = 230630 with 446 = 843^5 + 1771^5 - 177372^3 - 230630^3.
a(769) = 614692 with 769 = 2320^5 + 2777^5 - 46750^3 - 614692^3.
(End)

Links

Crossrefs

Cf. A000578, A000584, A004842, A004999, A351338, A351341, A351376.

Programs

  • Mathematica
    FQ[n_]:=IntegerQ[n^(1/5)];
tab={};Do[m=0;Label[bb]; k=m^3;Do[If[FQ[n+k+x^3-y^5], tab=Append[tab,m];Goto[aa]],{x,0,m},{y,0,((n+k+x^3)/2)^(1/5)}];m=m+1;Goto[bb];Label[aa],{n,0,21}];Print[tab]

Extensions

a(22)-a(73) from Chai Wah Wu, Feb 21 2022

A351376 Least nonnegative integer m such that n = x^3 + y^3 - (z^5 + m^5) for some nonnegative integers x,y,z with z <= m.

Original entry on oeis.org

0, 0, 0, 2, 76, 3, 1, 1, 0, 0, 6, 5, 4, 7, 1, 1, 0, 51, 129, 14, 22, 2, 2, 4, 136, 1, 1, 0, 0, 27, 7, 2, 2, 1, 1, 0, 3, 3, 14, 2, 2, 44, 11, 5, 8, 6, 101, 4, 4, 28, 14, 6, 1, 1, 0, 17, 42, 33, 2, 2, 20, 2, 1, 1, 0, 0, 3, 8, 3, 2, 1, 1, 0, 3, 6, 41, 3, 43, 12, 10, 10, 6, 6, 6, 59, 29, 33, 81, 2, 1, 1, 0, 2, 2, 2, 2, 2, 3, 3, 3, 2
Offset: 0

Views

Author

Zhi-Wei Sun, Feb 09 2022

Keywords

Comments

Conjecture: a(n) exists for any nonnegative integer n.
See also Conjecture 1 in A351341.

Examples

			a(4) = 76 with 4 = 775^3 + 1397^3 - (58^5 + 76^5).
a(18) = 129 with 18 = 1693^3 + 3137^3 - (3^5 + 129^5).
a(24) = 136 with 24 = 2534^3 + 3116^3 - (0^5 + 136^5).
a(87) = 81 with 87 = 140^3 + 1658^3 - (64^5 + 81^5).
From _Chai Wah Wu_, Feb 21 2022 : (Start)
a(389) = 3883 with 389 = 590621^3 + 877987^3 - (612^5 + 3883^5).
a(4173) = 3978 with 4173 = 16112^3 + 1108958^3 - (3259^5 + 3978^5).
(End)

Links

Crossrefs

Cf. A000578, A000584, A004842, A004999, A351338, A351341, A351375.

Programs

  • Mathematica
    CQ[n_]:=IntegerQ[n^(1/3)];
tab={};Do[m=0;Label[bb]; k=m^5;Do[If[CQ[n+k+x^5-y^3], tab=Append[tab,m];Goto[aa]],{x,0,m},{y,0,((n+k+x^5)/2)^(1/3)}];m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]

A351832 Least nonnegative integer m such that n = x^6 + y^6 - (z^2 + m^2) for some nonnegative integers x,y,z with z <= m.

Original entry on oeis.org

0, 0, 0, 6, 6, 20, 7, 7, 19, 24, 24, 7, 6, 6, 5, 5, 7, 26, 26, 6, 6, 22, 9, 5, 5, 6, 98, 6, 6, 6, 5, 5, 4, 4, 32, 5, 5, 26, 5, 4, 4, 20, 322, 7, 4, 4, 3, 3, 4, 4, 22, 3, 3, 22, 3, 3, 2, 2, 418, 2, 2, 2, 1, 1, 0, 0, 94, 6, 23, 20, 7, 19, 24, 20, 20, 7, 6, 22, 5, 7, 19, 18, 18, 6, 22, 37, 59, 5, 6, 24, 24, 6, 6, 21
Offset: 0

Views

Author

Zhi-Wei Sun, Feb 21 2022

Keywords

Comments

Conjecture: a(n) exists for each nonnegative integer n.
See also Conjecture 2 in A351341.

Examples

			a(170) = 2730 with 170 = 9^6 + 15^6 - (2114^2 + 2730^2).
a(5938) = 16184 with 5938 = 17^6 + 25^6 - (2520^2 + 16184^2).
a(9746) = 7600 with 9746 = 11^6 + 21^6 - (5456^2 + 7600^2).

Links

Crossrefs

Cf. A000290, A001014, A001481, A351321, A351338, A351341, A351375, A351376.

Programs

  • Mathematica
    QQ[n_]:=IntegerQ[n^(1/6)];
tab={};Do[m=0; Label[bb]; k=m^2; Do[If[QQ[n+k+x^2-y^6], tab=Append[tab,m];Goto[aa]],  {x, 0, m}, {y, 0, ((n+k+x^2)/2)^(1/6)}];m=m+1; Goto[bb]; Label[aa], {n, 0, 100}];Print[tab]
Showing 1-5 of 5 results.

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