A351902 -id:A351902 - cp's OEIS frontend

A351723 Numbers of the form x^2 + y^2 + z^2 + xyz with x,y,z nonnegative integers.

0, 1, 2, 4, 5, 8, 9, 10, 13, 14, 16, 17, 18, 20, 22, 25, 26, 28, 29, 32, 34, 36, 37, 38, 40, 41, 44, 45, 49, 50, 52, 53, 54, 58, 61, 62, 64, 65, 68, 70, 72, 73, 74, 76, 77, 80, 81, 82, 85, 88, 89, 90, 92, 94, 97, 98, 100, 101, 104, 106, 108, 109, 110, 112, 113, 116, 117, 118, 121, 122, 125, 128, 130, 133, 134, 136

Offset: 1

Zhi-Wei Sun, Feb 17 2022

nonn

It is easy to see that no term is congruent to 3 modulo 4.
Conjecture 1: a(n) < 2*n for all n > 0, and a(n)/n has a limit as n tends to the infinity. Also, a(n) <= a(n-1) + a(n-2) for all n > 4.
Conjecture 2: Let S = {x^2 + y^2 + z^2 + x*y*z: x,y,z = 0,1,2,...}.
(i) 7 and 487 are the only nonnegative integers which cannot be written as w^2 + s, where w is a nonnegative integer and s is an element of S. Also, 7, 87 and 267 are the only nonnegative integers which cannot be written as w^3 + s, where w is a nonnegative integer and s is an element of S.
(ii) Let k be 2 or 3. Then each nonnegative integer not congruent to 3 modulo 4 can be written as 4*w^k + s, where w is a nonnegative integer and s is an element of S.
This has been verified for nonnegative integers up to 10^6.

			a(3) = 2 with 2 = 1^2 + 1^2 + 0^2 + 1*1*0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Can each natural number be represented by 2*w^2+x^2+y^2+z^2+xyz with x,y,z in {0,1,2,...}? Question 416344 at MathOverflow, Feb. 17, 2022.

Cf. A000290, A000378, A000578, A351902.

tab={};Do[n=x^2+y^2+z^2+x*y*z;If[n<=140,tab=Append[tab,n]],{x,0,20},{y,0,x},{z,0,y}];Print[Sort[DeleteDuplicates[tab]]]

A351617 Number of ways to write n as 11^w + x^2 + 2y^2 + 3z^2 + xyz, 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

Zhi-Wei Sun, Mar 10 2022

nonn

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.

			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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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

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]

A352259 Number of ways to write n as w^6 + x^2 + 2y^2 + 3z^2 + xyz, where w,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, 5, 6, 6, 10, 8, 3, 3, 4, 5, 8, 4, 9, 13, 12, 8, 2, 7, 10, 9, 10, 9, 7, 5, 3, 3, 8, 5, 10, 10, 6, 7, 8, 6, 10, 9, 11, 10

Offset: 0

Zhi-Wei Sun, Mar 10 2022

nonn

Conjecture 1: (i) a(n) > 0 for every n = 0,1,2,.... Moreover, 106, 744, 5469 and 331269 are the only nonnegative integers not in the set {w + x^2 + 2*y^2 + 3*z^2 + x*y*z: w = 0,1; x,y,z = 0,1,2,...}.
(ii) Let k be one of 4, 5, 6, 7. Then each n = 0,1,2,... can be written as 10*w^k + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
(iii) Let c be among 1, 3, 4, 6, 7, and let k be 4 or 5. Then every n = 0,1,2,... can be written as c*w^k + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
(iv) Each n = 0,1,2,... can be written as 9*w^4 + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
Conjecture 2: Every n = 0,1,2,... can be written as 2*w^4 + 3*x^2 + y^2 + z^2 + x*y*z, where w,x,y,z are nonnegative integers.
We have verified Conjectures 1 and 2 for all n <= 10^5.

			a(24) = 1 with 24 = 0^6 + 4^2 + 2*2^2 + 3*0^2 + 4*2*0.
a(106) = 1 with 106 = 2^6 + 1^2 + 2*2^2 + 3*3^2 + 1*2*3.

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

Cf. A000290, A000583, A000584, A001014, A351723, A351617, A351902, A352286.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[4(n-w^6-2y^2-3z^2)+y^2*z^2],r=r+1],{w,0,n^(1/6)},{z,0,Sqrt[(n-w^6)/3]},{y,0,Sqrt[(n-w^6-3z^2)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]

A352286 Number of ways to write n as w + x^2 + 2y^2 + 3z^2 + xyz, 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

Zhi-Wei Sun, Mar 10 2022

nonn

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.

			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.

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

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

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 12n + 5 as 2x^2 + 5y^2 + 9z^2 + xyz, 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

Zhi-Wei Sun, Mar 13 2022

nonn

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.

			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.

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

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

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]

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A351723 Numbers of the form x^2 + y^2 + z^2 + x*y*z with x,y,z nonnegative integers.

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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.

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A352259 Number of ways to write n as w^6 + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

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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.

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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.

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A351723 Numbers of the form x^2 + y^2 + z^2 + xyz with x,y,z nonnegative integers.

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

A352259 Number of ways to write n as w^6 + x^2 + 2y^2 + 3z^2 + xyz, where w,x,y,z are nonnegative integers.

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

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