A280356 -id:A280356 - cp's OEIS frontend

A350714 Least positive integer m such that m^12*n = x^4 + y^3 + z^2 for some nonnegative integers x,y,z.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1

Offset: 0

Zhi-Wei Sun, Feb 02 2022

nonn

4-3-2 Conjecture: a(n) exists for any nonnegative integer n. Equivalently, each nonnegative rational number can be written x^4 + y^3 + z^2 with x,y,z nonnegative rational numbers.
Note that m/n = (m*n^11)/n^12 for any positive integers m and n.
a(n) <= 4 for n <= 40000 with the only exception a(23710) = 5.
a(n) <= 4 for n = 77000..100000, and a(n) = 4 for n = 78367, 79479, 83494, 84694, 85979, 86822, 87395, 87814, 90047, 90278, 92891, 93715.
Qing-Hu Hou verified a(n) <= 4 for 40000 < n < 77000. - Zhi-Wei Sun Feb 04 2022
a(n) <= 5 for 10^5 < n <= 2*10^5, and a(n)=5 for n=107206, 117615, and 148079. - Qing-Hu Hou, Feb 05 2022

			a(6) = 1 with 1^12*6 = 1^4 + 1^3 + 2^2.
a(7) = 2 with 2^12*7 = 2^4 + 15^3 + 159^2.
a(75) = 4 with 4^12*75 = 122^4 + 1007^3 + 3951^2.
a(1140) = 3 with 3^12*1140 = 0^4 + 531^3 + 21357^2.
a(23710) = 5 with 5^12*23710 = 217^4 + 17897^3 + 232166^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000290, A000578, A000583, A280356.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[m=1;Label[bb];k=m^12;Do[If[SQ[k*n-x^4-y^3],tab=Append[tab,m];Goto[aa]],{x,0,(k*n)^(1/4)},{y,0,(k*n-x^4)^(1/3)}];m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]

A280153 Number of ways to write n as x^3 + 2*y^3 + z^2 + 4^k, where x is a positive integer and y,z,k are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 27 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 7, 10, 17, 24, 36, 38, 52, 62, 115, 136, 185, 990.
(ii) Each positive integer n can be written as x^2 + 2*y^2 + z^3 + 8^k with x,y,z,k nonnegative integers.
(iii) Let a,b,c be positive integers with b <= c. Then any positive integer can be written as a*x^3 + b*y^2 + c*z^2 + 4^k with x,y,z,k nonnegative integers, if and only if (a,b,c) is among the following triples: (1,1,1), (1,1,2), (1,1,3), (1,1,5), (1,1,6), (1,2,3), (1,2,5), (2,1,1), (2,1,2), (2,1,3), (2,1,6), (4,1,2), (5,1,2), (8,1,2), (9,1,2).
We have verified that a(n) > 0 for all n = 2..10^6, and that part (ii) of the conjecture holds for all n = 1..10^6.
For any positive integer n, it is easy to see that at least one of n-1, n-8, n-64 is not of the form 4^k*(8m+7) with k and m nonnegative integers, thus, by the Gauss-Legendre theorem on sums of three squares, n = x^2 + y^2 + z^2 + 8^k for some nonnegative integers x,y,z and k < 3.
See also A280356 for a similar conjecture involving powers of two.

			a(7) = 1 since 7 = 1^3 + 2*1^3 + 0^2 + 4^1.
a(10) = 1 since 10 = 2^3 + 2*0^3 + 1^2 + 4^0.
a(17) = 1 since 17 = 1^3 + 2*0^3 + 0^2 + 4^2.
a(24) = 1 since 24 = 2^3 + 2*0^3 + 0^2 + 4^2.
a(36) = 1 since 36 = 2^3 + 2*1^3 + 5^2 + 4^0.
a(38) = 1 since 38 = 1^3 + 2*0^3 + 6^2 + 4^0.
a(52) = 1 since 52 = 3^3 + 2*0^3 + 3^2 + 4^2.
a(62) = 1 since 62 = 2^3 + 2*1^3 + 6^2 + 4^2.
a(115) = 1 since 115 = 2^3 + 2*3^3 + 7^2 + 4^1.
a(136) = 1 since 136 = 2^3 + 2*0^3 + 8^2 + 4^3.
a(185) = 1 since 185 = 3^3 + 2*3^3 + 10^2 + 4^1.
a(990) = 1 since 990 = 7^3 + 2*3^3 + 23^2 + 4^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000290, A000302, A000578, A262813, A262827, A262857, A270533, A270559, A280356.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-4^k-x^3-2y^3],r=r+1],{k,0,Log[4,n]},{x,1,(n-4^k)^(1/3)},{y,0,((n-4^k-x^3)/2)^(1/3)}];Print[n," ",r];Continue,{n,1,80}]

A282426 Number of ways to write n as x^4 + 4*y^2 + z^2 + 3^k, where x,y,z are nonnegative integers and k is among 0,1,2,3,4.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 2, 3, 4, 2, 1, 5, 6, 5, 4, 5, 4, 5, 3, 4, 7, 5, 5, 7, 5, 3, 5, 6, 5, 6, 4, 5, 5, 6, 4, 7, 7, 6, 9, 6, 5, 6, 5, 4, 6, 4, 4, 7, 6, 6, 6, 5, 2, 6, 3, 4, 6, 6, 5, 6, 6, 6, 10, 7, 5

Offset: 1

Zhi-Wei Sun, Feb 14 2017

nonn

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) Each n = 1,2,3,... can be written as x^4 + y^2 + z^2 + 2^k with x,y,z nonnegative integers and k among 0,1,2,3,4, 5.
(iii) Any positive integer n can be written as 4*x^4 + y^3 + z^2 + 3^k with k,x,y,z nonnegative integers.
We have verified parts (i) and (ii)-(iii) for n up to 2*10^7 and 10^7 respectively.
See arXiv:1701.05868 for more such conjectures.

			a(1) = 1 since 1 = 0^4 + 4*0^2 + 0^2 + 3^0.
a(16) = 1 since 16 = 0^4 + 4*1^2 + 3^2 + 3^1.
a(475) = 1 since 475 = 3^4 + 4*6^2 + 13^2 + 3^4.
a(556) = 1 since 556 = 0^4 + 4*0^2 + 23^2 + 3^3.
a(8641) = 1 since 8641 = 9^4 + 4*21^2 + 17^2 + 3^3.
a(52696) = 1 since 52696 = 12^4 + 4*87^2 + 41^2 + 3^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000079, A000244, A000290, A000583, A280356.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-4y^2-3^k],r=r+1],{k,0,Min[4,Log[3,n]]},{x,0,(n-3^k)^(1/4)},{y,0,Sqrt[(n-3^k-x^4)/4]}];Print[n," ",r];Continue,{n,1,80}]

A297995 Number of ways to write n as 4u + v^2 + x^3 + y^4 + 2z^8, where u is 0 or 1, v is a positive integer and x,y,z are nonnegative integers.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 6, 5, 5, 5, 7, 6, 6, 5, 4, 5, 5, 6, 3, 5, 4, 4, 5, 5, 5, 5, 6, 5, 5, 5, 4, 4, 4, 2, 2, 2, 3, 4, 6, 6, 5, 7, 6, 7, 5, 6, 4, 4, 4, 3, 4, 3, 5, 4

Offset: 1

Zhi-Wei Sun, Jan 10 2018

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 398, 496, 498, 500, 1507, 6419, 7843, 7983, 8688, 8947, 9175, 9251, 12923, 12976, 48381.

(ii) For any positive integers a,b,c,d,e and integers g,h,i,j,k greater than one, if each n = 0,1,2,... can be written as a*u^g + b*v^h + c*x^i + d*y^j + e*z^k with u,v,x,y,z nonnegative integers, then 1/g + 1/h + 1/i + 1/j + 1/k is greater than 1/2 + 1/3 + 1/4 + 1/8 = 29/24.

			a(500) = 1 since 500 = 4*0 + 22^2 + 0^3 + 2^4 + 2*0^8.
a(1507) = 1 since 1507 = 4*1 + 13^2 + 11^3 + 1^4 + 2*1^8.
a(9251) = 1 since 9251 = 4*0 + 91^2 + 7^3 + 5^4 + 2*1^8.
a(12923) = 1 since 12923 = 4*1 + 54^2 + 1^3 + 10^4 + 2*1^8.
a(12976) = 1 since 12976 = 4*0 + 92^2 + 15^3 + 5^4 + 2*2^8.
a(48381) = 1 since 48381 = 4*1 + 140^2 + 6^3 + 13^4 + 2*0^8.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. (See Remark 3.5.)

Cf. A000290, A000578, A000583, A262827, A280356.

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-4u-2z^8-y^4-x^3],r=r+1],{u,0,Boole[n>3]},{z,0,((n-4u)/2)^(1/8)},{y,0,(n-4u-2z^8)^(1/4)},{x,0,(n-4u-2z^8-y^4)^(1/3)}];Print[n," ",r],{n,1,80}]

cp's OEIS Frontend

A350714 Least positive integer m such that m^12*n = x^4 + y^3 + z^2 for some nonnegative integers x,y,z.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A280153 Number of ways to write n as x^3 + 2*y^3 + z^2 + 4^k, where x is a positive integer and y,z,k are nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A282426 Number of ways to write n as x^4 + 4*y^2 + z^2 + 3^k, where x,y,z are nonnegative integers and k is among 0,1,2,3,4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A297995 Number of ways to write n as 4*u + v^2 + x^3 + y^4 + 2*z^8, where u is 0 or 1, v is a positive integer and x,y,z are nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A297995 Number of ways to write n as 4u + v^2 + x^3 + y^4 + 2z^8, where u is 0 or 1, v is a positive integer and x,y,z are nonnegative integers.