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 1-7 of 7 results.

A343397 Number of ways to write n as 2^x + [y^2/3] + [z^2/4] with x,y,z positive integers, where [.] is the floor function.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 5, 8, 5, 9, 5, 8, 8, 6, 9, 9, 10, 8, 11, 10, 10, 9, 9, 14, 8, 8, 10, 12, 11, 6, 14, 13, 10, 12, 13, 15, 11, 13, 9, 20, 6, 12, 17, 13, 13, 10, 11, 17, 12, 11, 13, 15, 14, 9, 13, 13, 14, 11, 18, 11, 15, 7, 12, 22, 13, 14, 17, 17, 11, 15, 13, 24, 16, 9, 17, 15, 15, 14, 18
Offset: 1

Views

Author

Zhi-Wei Sun, Apr 13 2021

Keywords

Comments

Conjecture: a(n) > 0 for all n > 1.
We have verified a(n) > 0 for all n = 2..2*10^6.
Conjecture verified up to 10^10. - Giovanni Resta, Apr 14 2021

Examples

			a(2) = 1 with 2 = 2^1 + [1^2/3] + [1^2/4].
a(3) = 2 with 3 = 2^1 + [1^2/3] + [2^2/4] = 2^1 + [2^2/3] + [1^2/4].

Links

Crossrefs

Cf. A000079, A000290, A343326, A343368, A343384, A343387, A343391, A343411.

Programs

  • Mathematica
    PowQ[n_]:=PowQ[n]=n>1&&IntegerQ[Log[2,n]];
tab={};Do[r=0;Do[If[PowQ[n-Floor[x^2/3]-Floor[y^2/4]],r=r+1],{x,1,Sqrt[3n-1]},{y,1,Sqrt[4(n-Floor[x^2/3]-1)+1]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A343391 Number of ways to write n as x^2 + [y^2/4] + [z^4/6] with x,y,z positive integers, where [.] is the floor function.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Apr 13 2021

Keywords

Comments

Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..10^6.
See also A343387 for a similar conjecture.

Examples

			a(1) = 1 with 1 = 1^2 + [1^2/4] + [1^4/6].
a(2) = 1 with 2 = 1^2 + [2^2/4] + [1^4/6].
a(14) = 1 with 14 = 1^2 + [1^2/4] + [3^4/6].
a(32) = 1 with 32 = 4^2 + [8^2/4] + [1^4/6].
a(35) = 1 with 35 = 4^2 + [5^2/4] + [3^4/6].
a(77) = 1 with 77 = 8^2 + [1^2/4] + [3^4/6].
a(840) = 1 with 840 = 28^2 + [15^2/4] + [1^4/6].

Links

Crossrefs

Cf. A000290, A000583, A343326, A343368, A343384, A343387, A343397, A343411.

Programs

  • Mathematica
    SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-Floor[x^2/4]-Floor[y^4/6]],r=r+1],{x,1,Sqrt[4n+3]},{y,1,(6(n-Floor[x^2/4])+5)^(1/4)}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343411 Number of ways to write n as x^3 + [y^3/2] + [z^3/3] + 2^w, where [.] is the floor function, x,y,z are positive integers, and w is a nonnegative integer.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 3, 2, 3, 4, 5, 2, 7, 2, 2, 2, 5, 4, 5, 5, 3, 3, 3, 3, 7, 6, 3, 5, 5, 6, 2, 11, 3, 6, 2, 6, 6, 8, 10, 2, 9, 2, 5, 5, 10, 5, 2, 6, 4, 4, 7, 5, 7, 2, 2, 4, 6, 7, 3, 12, 3, 7, 4, 9, 6, 5, 10, 4, 15, 4, 8, 5, 11, 4, 8, 14, 6, 4, 6, 10, 7, 8, 9, 5, 6, 4, 4, 13, 5, 7, 3, 10, 2, 7, 11
Offset: 1

Views

Author

Zhi-Wei Sun, Apr 14 2021

Keywords

Comments

Conjecture: a(n) > 0 for all n > 1.
We have verified a(n) > 0 for all 1 < n <= 3*10^5.
Conjecture verified up to 10^10. - Giovanni Resta, Apr 14 2021

Examples

			a(2) = 1 with 2 = 1^3 + [1^3/2] + [1^3/3] + 2^0.
a(3) = 1 with 3 = 1^3 + [1^3/2] + [1^3/3] + 2^1.
a(4) = 1 with 4 = 1^3 + [1^3/2] + [2^3/3] + 2^0.
a(6) = 1 with 6 = 1^3 + [2^3/2] + [1^3/3] + 2^0.
a(8) = 1 with 8 = 1^3 + [2^3/2] + [2^3/3] + 2^0.
a(10) = 1 with 10 = 2^3 + [1^3/2] + [1^3/3] + 2^1.
a(103) = 1 with 103 = 3^3 + [1^3/2] + [6^3/3] + 2^2.

Links

Crossrefs

Cf. A000079, A000578, A343326, A343368, A343384, A343387, A343391, A343397.

Programs

  • Mathematica
    CQ[n_]:=CQ[n]=n>0&&IntegerQ[n^(1/3)];
tab={};Do[r=0;Do[If[CQ[n-Floor[x^3/2]-Floor[y^3/3]-2^z],r=r+1],{x,1,(2n-1)^(1/3)},{y,1,(3(n-Floor[x^3/2])-1)^(1/3)},{z,0,Log[2,n-Floor[x^3/2]-Floor[y^3/3]]}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343384 Number of ways to write n as [a^3/3] + [b^3/4] + [c^3/5] + [d^6/6] with a,b,c,d positive integers, where [.] is the floor function.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Apr 13 2021

Keywords

Comments

3-4-5-6 Conjecture: a(n) > 0 for all n >= 0.
We have verified a(n) > 0 for all n = 0..10^6.
Conjecture verified up to 2*10^9. - Giovanni Resta, Apr 28 2021

Examples

			a(0) = 1 with 0 = [1^3/3] + [1^3/4] + [1^3/5] + [1^6/6].
a(1) = 1 with 1 = [1^3/3] + [1^3/4] + [2^3/5] + [1^6/6].
a(4) = 1 with 4 = [2^3/3] + [2^3/4] + [1^3/5] + [1^6/6].
a(6) = 1 with 6 = [1^3/3] + [3^3/4] + [1^3/5] + [1^6/6].
a(8) = 1 with 8 = [2^3/3] + [3^3/4] + [1^3/5] + [1^6/6].
a(60) = 1 with 60 = [3^3/3] + [4^3/4] + [5^3/5] + [2^6/6].
a(81) = 1 with 81 = [2^3/3] + [6^3/4] + [5^3/5] + [1^6/6].
a(300) = 1 with 300 = [7^3/3] + [5^3/4] + [9^3/5] + [2^6/6].
a(4434) = 1 with 4434 = [11^3/3] + [4^3/4] + [19^3/5] + [5^6/6].

Links

Crossrefs

Cf. A000578, A001014, A343326, A343368, A343387.

Programs

  • Mathematica
    CQ[n_]:=CQ[n]=n>0&&IntegerQ[n^(1/3)];
tab={};Do[r=0;Do[If[CQ[3(n-Floor[x^6/6]-Floor[y^3/5]-Floor[z^3/4])+s],r=r+1],{s,0,2},{x,1,(6n+5)^(1/6)},{y,1,(5(n-Floor[x^6/6])+4)^(1/3)},{z,1,(4(n-Floor[x^6/6]-Floor[y^3/5])+3)^(1/3)}];tab=Append[tab,r],{n,0,100}];Print[tab]

A338686 Number of ways to write n as x^5 + y^2 + [z^2/7], where x,y,z are integers with x >= 0, y >= 1 and z >= 2, and [.] is the floor function.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 2, 3, 4, 4, 4, 3, 2, 3, 4, 3, 5, 5, 2, 4, 3, 2, 3, 4, 5, 4, 4, 4, 4, 2, 2, 6, 5, 2, 6, 8, 6, 4, 5, 6, 6, 5, 4, 6, 5, 4, 5, 4, 10, 6, 5, 8, 3, 5, 5, 7, 6, 4, 5, 7, 5, 2, 6, 7, 6, 7, 8, 6, 4, 5, 6, 8, 6, 2, 4, 8, 4, 6, 3, 5, 10, 6, 8, 7, 5, 5, 6, 5, 5, 5, 7, 6, 4, 4, 6, 3, 8, 7, 7
Offset: 1

Views

Author

Zhi-Wei Sun, Apr 23 2021

Keywords

Comments

Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..7*10^6.
See also A338687 for a similar conjecture.
Conjecture verified up to 2*10^9. - Giovanni Resta, Apr 28 2021

Examples

			a(1) = 1 with 1 = 0^5 + 1^2 + [2^2/7].
a(166) = 1 with 166 = 0^5 + 1^2 + [34^2/7].
a(323) = 1 with 323 = 2^5 + 17^2 + [4^2/7].
a(815) = 1 with 815 = 2^5 + 1^2 + [74^2/7].
a(2069) = 1 with 2069 = 0^5 + 37^2 + [70^2/7].
a(7560) = 1 with 7560 = 2^5 + 64^2 + [155^2/7].
a(24195) = 1 with 24195 = 0^5 + 8^2 + [411^2/7].
a(90886) = 2 with 90886 = 4^5 + 34^2 + [788^2/7] = 9^5 + 139^2 + [296^2/7].

Links

Crossrefs

Cf. A000290, A000584, A270920, A338687, A343387, A343391, A343397.

Programs

  • Mathematica
    SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-x^5-Floor[y^2/7]],r=r+1],{x,0,(n-1)^(1/5)},{y,2,Sqrt[7(n-x^5)-1]}];tab=Append[tab,r],{n,1,100}];Print[tab]

A338687 Number of ways to write n as x^4 + y^2 + floor(z^2/7), where x,y,z are integers with x >= 0, y >= 1 and z >= 2.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 2, 3, 4, 4, 4, 3, 2, 3, 4, 4, 6, 6, 4, 5, 5, 3, 4, 6, 7, 6, 6, 5, 5, 4, 4, 6, 8, 2, 5, 10, 4, 5, 5, 7, 6, 5, 4, 7, 6, 2, 5, 6, 7, 5, 8, 8, 4, 5, 6, 6, 6, 3, 4, 9, 3, 4, 5, 6, 9, 8, 7, 5, 4, 5, 6, 8, 6, 1, 6, 6, 5, 5, 5, 4, 11, 9, 7, 9, 6, 7, 7, 8, 5, 8, 8, 8, 6, 6, 5, 7, 8, 10, 10
Offset: 1

Views

Author

Zhi-Wei Sun, Apr 23 2021

Keywords

Comments

Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..5*10^6.
See also A338686 for a similar conjecture.

Examples

			a(1) = 1 with 1 = 0^4 + 1^2 + floor(2^2/7).
a(75) = 1 with 75 = 0^4 + 8^2 + floor(9^2/7).
a(1799) = 1 with 1799 = 5^4 + 25^2 + floor(62^2/7).
a(7224) = 1 with 7224 = 9^4 + 19^2 + floor(46^2/7).
a(27455) = 2 with 27455 = 0^4 + 7^2 + floor(438^2/7) = 8^4 + 118^2 + floor(257^2/7).

Links

Crossrefs

Cf. A000290, A000583, A270566, A338686, A338696, A343387, A343391, A343397.

Programs

  • Mathematica
    SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-x^4-Floor[y^2/7]],r=r+1],{x,0,(n-1)^(1/4)},{y,2,Sqrt[7(n-x^4)-1]}];tab=Append[tab,r],{n,1,100}];Print[tab]

A343400 Number of ways to write n as 3^x + [y^2/3] + [z^2/4], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 4, 6, 4, 9, 5, 7, 8, 6, 9, 6, 7, 9, 7, 6, 9, 7, 8, 7, 7, 10, 6, 9, 11, 9, 12, 8, 9, 14, 5, 13, 11, 8, 11, 11, 7, 13, 9, 12, 11, 9, 9, 11, 8, 12, 11, 11, 11, 6, 16, 4, 11, 12, 11, 13, 12, 6, 10, 9, 8, 17, 8, 12, 11, 10, 8, 10, 12, 10, 8, 11, 12, 12, 13, 7
Offset: 1

Views

Author

Zhi-Wei Sun, Apr 13 2021

Keywords

Comments

Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..2*10^6.
The first indices n for which a(n) = 0 are 4051736, 7479656, 8592680, 9712160, 14039792, 16726256, 24914510. - Giovanni Resta, Apr 14 2021

Examples

			a(2) = 2 with 2 = 3^0 + [1^2/3] + [2^2/4] = 3^0 + [2^2/3] + [1^2/4].
a(2942) = 2 with 2942 = 3^1 + [93^2/3] + [15^2/4] = 3^7 + [44^2/3] + [21^2/4].
a(627662) = 5 with 627662 - 3^0 = [330^2/3] + [1538^2/4] = [1042^2/3] + [1031^2/4] = [1318^2/3] + [441^2/4] = [1328^2/3] + [399^2/4] = [1352^2/3] + [271^2/4].
a(1103096) = 3 with 1103096 = 3^1 + [260^2/3] + [2079^2/4] = 3^1 + [508^2/3] + [2017^2/4] = 3^9 + [328^2/3] + [2047^2/4].
a(1694294) = 3 with 1694294 = 3^8 + [860^2/3] + [2401^2/4] = 3^8 + [928^2/3] + [2367^2/4] = 3^13 + [112^2/3] + [619^2/4].

Links

Crossrefs

Cf. A000290, A000244, A343387, A343391, A343397.

Programs

  • Mathematica
    PowQ[n_]:=PowQ[n]=IntegerQ[Log[3,n]];
tab={};Do[r=0;Do[If[PowQ[n-Floor[x^2/3]-Floor[y^2/4]],r=r+1],{x,1,Sqrt[3n-1]},{y,1,Sqrt[4(n-Floor[x^2/3]-1)+1]}];tab=Append[tab,r],{n,1,80}];Print[tab]
Showing 1-7 of 7 results.

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