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 1-3 of 3 results.

A351617 Number of ways to write n as 11^w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Mar 10 2022

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) Let c be among 3, 4, 5, 7, 8. Then each positive integer n can be written as c^w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
This has been verified for all n = 1..3*10^5.

Examples

			a(6) = 1 with 6 = 11^0 + 0^2 + 2*1^2 + 3*1^2 + 0*1*1.
a(24) = 1 with 24 = 11^1 + 1^2 + 2*0^2 + 3*2^2 + 1*0*2.
a(71) = 1 with 71 = 11^0 + 4^2 + 2*3^2 + 3*2^2 + 4*3*2.
a(89) = 1 with 89 = 11^0 + 4^2 + 2*6^2 + 3*0^2 + 4*6*0.
a(107) = 1 with 107 = 11^1 + 8^2 + 2*4^2 + 3*0^2 + 8*4*0.

Links

Crossrefs

Cf. A000290, A001014, A351723, A351902, A352259, A352286.

Programs

  • Mathematica
    SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[4(n-11^w-2y^2-3z^2)+y^2*z^2],r=r+1],{w,0,Log[11,n]},{z,0,Sqrt[(n-11^w)/3]},{y,0,Sqrt[(n-11^w-3z^2)/2]}];tab=Append[tab,r],{n,1,100}];Print[tab]

A352286 Number of ways to write n as w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w is 0 or 1, and x,y,z are nonnegative integers.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 2, 3, 3, 3, 2, 3, 6, 4, 3, 2, 2, 5, 5, 5, 4, 3, 4, 2, 1, 5, 5, 4, 6, 5, 3, 3, 4, 5, 4, 5, 7, 5, 4, 5, 4, 3, 3, 3, 4, 3, 3, 5, 6, 7, 6, 5, 7, 6, 4, 4, 4, 7, 5, 4, 4, 3, 7, 5, 4, 5, 5, 8, 6, 2, 2, 2, 4, 6, 4, 6, 10, 11, 6, 2, 5, 7, 7, 7, 8, 5, 3, 3, 2, 4, 4, 7, 7, 4, 6, 6, 4, 7, 8, 7, 7
Offset: 0

Views

Author

Zhi-Wei Sun, Mar 10 2022

Keywords

Comments

Conjecture 1: a(n) = 0 only for n = 106, 744, 5469, 331269. Thus, for any nonnegative integer n not among 106, 744, 5469 and 331269, either n or n - 1 can be written as x^2 + 2*y^2 + 3*z^2 + x*y*z with x,y,z nonnegative integers.
Conjecture 2: a(n) = 1 only for n = 0, 24, 346, 360, 664, 667, 1725, 2589, 3111, 4906, 5035, 8043, 8709, 16810, 18699, 34539, 39256, 51621, 59019, 62799, 108645, 136167, 562696.
We have verified Conjectures 1 and 2 for n = 0..10^6.

Examples

			a(24) = 1 with 24 = 0 + 4^2 + 2*2^2 + 3*0^2 + 4*2*0.
a(346) = 1 with 346 = 1 + 15^2 + 2*3^2 + 3*2^2 + 15*3*2.
a(360) = 1 with 360 = 1 + 9^2 + 2*5^2 + 3*4^2 + 9*5*4.
a(62799) = 1 with 62799 = 1 + 16^2 + 2*169^2 + 3*2^2 + 16*169*2.
a(108645) = 1 with 108645 = 0 + 95^2 + 2*163^2 + 3*3^2 + 95*163*3.
a(136167) = 1 with 136167 = 0 + 2^2 + 2*17^2 + 3*207^2 + 2*17*207.
a(562696) = 1 with 562696 = 0 + 539^2 + 2*20^2 + 3*25^2 + 539*20*25.

Links

Crossrefs

Cf. A000290, A351617, A351723, A351902, A352259.

Programs

  • Mathematica
    SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[4(n-w-2y^2-3z^2)+y^2*z^2],r=r+1],{w,0,Min[1,n]},{z,0,Sqrt[(n-w)/3]},{y,0,Sqrt[(n-w-3z^2)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]

A352356 Number of ways to write 12*n + 5 as 2*x^2 + 5*y^2 + 9*z^2 + x*y*z, where x, y and z are nonnegative integers.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 2, 4, 3, 3, 2, 5, 1, 3, 4, 3, 3, 7, 4, 2, 5, 3, 4, 3, 5, 5, 5, 6, 4, 4, 4, 2, 4, 5, 6, 3, 6, 5, 6, 5, 4, 5, 6, 7, 4, 4, 6, 4, 7, 6, 5, 3, 3, 8, 3, 7, 7, 4, 5, 7, 5, 6, 6, 8, 4, 1, 4, 7, 4, 8, 6, 5, 8, 9, 8, 4, 8, 3, 7, 4, 4, 12, 3, 4, 11, 8, 1, 6, 7, 5, 5, 8, 9, 5, 8, 12, 5, 6, 6, 6, 6
Offset: 0

Views

Author

Zhi-Wei Sun, Mar 13 2022

Keywords

Comments

Conjecture: For each n = 0,1,2,... we can write 12*n + 5 as 2*x^2 + 5*y^2 + 9*z^2 + x*y*z with x,y,z nonnegative integers.
This has been verified for all n = 0..10^6.
It seems that a(n) = 1 only for n = 0, 2, 5, 12, 65, 86, 155, 338, 21030.

Examples

			a(0) = 1 with 12*0 + 5 = 5 = 2*0^2 + 5*1^2 + 9*0^2 + 0*1*0.
a(2) = 1 with 12*2 + 5 = 29 = 2*0^2 + 5*2^2 + 9*1^2 + 0*2*1.
a(5) = 1 with 12*5 + 5 = 65 = 2*3^2 + 5*1^2 + 9*2^2 + 3*1*2.
a(12) = 1 with 12*12 + 5 = 149 = 2*0^2 + 5*1^2 + 9*4^2 + 0*1*4.
a(65) = 1 with 12*65 + 5 = 785 = 2*1^2 + 5*9^2 + 9*6^2 + 1*9*6.
a(86) = 1 with 12*86 + 5 = 1037 = 2*6^2 + 5*1^2 + 9*10^2 + 6*1*10.
a(155) = 1 with 12*155 + 5 = 1865 = 2*2^2 + 5*6^2 + 9*13^2 + 2*6*13.
a(338) = 1 with 12*338 + 5 = 4061 = 2*20^2 + 5*6^2 + 9*13^2 + 20*6*13.
a(21030) = 1 with 12*21030 + 5 = 252365 = 2*32^2 + 5*126^2 + 9*39^2 + 32*126*39.

Links

Crossrefs

Cf. A000290, A351723, A351617, A351902, A352259, A352286.

Programs

  • Mathematica
    SQ[n_]:=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[8(12n+5-5y^2-9z^2)+y^2*z^2]&&Mod[Sqrt[8(12n+5-5y^2-9z^2)+y^2*z^2]-y*z,4]==0,r=r+1],{y,0,Sqrt[(12n+5)/5]},{z,0,Sqrt[(12n+5-5y^2)/9]}];tab=Append[tab,r],{n,0,100}];Print[tab]
Showing 1-3 of 3 results.

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