A350714 -id:A350714 - cp's OEIS frontend

A351179 Least positive integer m such that m^6*n = w^6 + x^3 + y^3 + z^3 for some nonnegative integers w,x,y,z.

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

Offset: 0

Zhi-Wei Sun, Feb 04 2022

nonn

a(n) always exists, because any positive rational number can be written as a sum of three cubes of positive rational numbers (see Richmond reference).

Aside from a(96) = 7 and a(850) = 8, a(n) <= 6 for n <= 10^6. - Charles R Greathouse IV, Feb 10 2022

			a(5) = 3 with 3^6*5 = 2^6 + 5^3 + 12^3 + 12^3.
a(12) = 5 with 5^6*12 = 3^6 + 19^3 + 34^3 + 52^3.
a(22) = 2 with 2^6*22 = 1^6 + 4^3 + 7^3 + 10^3.
a(31) = 6 with 6^6*31 = 0^6 + 4^3 + 15^3 + 113^3.
a(96) = 7 with 7^6*96 = 0^6 + 2^3 + 38^3 + 224^3.
a(101) = 4 with 4^6*101 = 3^6 + 22^3 + 39^3 + 70^3.
a(850) = 8 with 8^6*850 = 5^6 + 508^3 + 442^3 + 175^3.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th Edition, Oxford Univ. Press, 1960. (See Theorem 234 on page 197.)

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
H. W. Richmond, On analogues of Waring's problem for rational numbers, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 401-409.
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. A000578, A001014, A004825, A350714, A351199.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={};Do[m=1;Label[bb];k=m^6;Do[If[CQ[k*n-w^6-x^3-y^3],tab=Append[tab,m];Goto[aa]],{w,0,(k*n)^(1/6)},{x,0,((k*n-w^6)/3)^(1/3)},{y,x,((k*n-w^6-x^3)/2)^(1/3)}];
m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]

a(n) <= A351199(n)^2. - Charles R Greathouse IV, Feb 05 2022

A351199 Least positive integer m such that m^3*n = x^3 + y^3 + z^3 for some nonnegative integers x,y,z.

Original entry on oeis.org

1, 1, 1, 1, 15, 6, 5, 3, 1, 1, 1, 11, 39, 3, 3, 3, 1, 1, 4, 2, 2, 3, 18, 6, 1, 22, 28, 1, 1, 1, 29, 15, 15, 21, 3, 1, 1, 7, 7, 25, 3, 12, 6, 1, 2, 7, 2, 7, 5, 21, 6, 2, 25, 5, 1, 1, 3, 3, 45, 132, 6, 45, 1, 3, 1, 1, 1, 171, 6, 9, 2, 3, 1, 1, 54, 21, 18, 3, 13, 32, 1, 1, 10, 2, 7, 9, 3, 3, 6, 3, 11, 1, 1, 63, 3, 30, 21, 5, 4, 1, 12

Offset: 0

Zhi-Wei Sun, Feb 04 2022

nonn

a(n) always exists, because any positive rational number can be written as a sum of three cubes of positive rational numbers (see Richmond reference).

			a(4) = 15 with 15^3*4 = 12^3 + 17^3 + 19^3.
a(212) = 216 with 216^3*212 = 82^3 + 161^3 + 1287^3.
a(446) = 228 with 228^3*446 = 929^3 + 1287^3 + 1330^3.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th Edition, Oxford Univ. Press, 1960. (See Theorem 234 on page 197.)

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms 0..600 from Zhi-Wei Sun)
H. W. Richmond, On analogues of Waring's problem for rational numbers, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 401-409.
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. A000578, A350714, A351179.

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

T=thueinit('x^3+1);
has2(n)=n==0 || #select(v->min(v[1], v[2])>=0, thue(T, n))>0
has3(n)=forstep(k=sqrtnint(n,3),sqrtnint(n\3,3),-1,if(has2(n-k^3),return(1)));0
a(n)=my(m=1); while(!has3(m^3*n), m++); m \\ Charles R Greathouse IV, Feb 05 2022

a(n) >= sqrt(A351179(n)).

A351206 Least positive integer m such that n = x^4 + (y^4 + z^4 + 7*w^2)/m^4 for some nonnegative integers x,y,z,w with y <= z.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 4, 3, 1, 1, 1, 6, 2, 2, 10, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 4, 2, 4, 10, 1, 1, 2, 4, 2, 1, 1, 1, 3, 1, 3, 2, 2, 2, 2, 2, 1, 3, 4, 2, 2, 1, 1, 3, 1, 1, 1, 1, 2, 4, 2, 2, 3, 8, 10, 2, 2, 1, 10, 5, 1, 1, 1, 1, 1, 4, 2, 2, 8, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 4, 4

Offset: 0

Zhi-Wei Sun, Feb 04 2022

nonn

Conjecture: a(n) exists for any nonnegative integer n.

This implies that each nonnegative rational number can be written as 7*w^2 + x^4 + y^4 + z^4 with w,x,y,z rational numbers.

			a(6) = 2 with 6 = 1^4 + (1^4 + 2^4 + 7*3^2)/2^4.
a(19) = 6 with 19 = 0^4 + (1^4 + 4^4 + 7*59^2)/6^4.
a(22) = 10 with 22 = 2^4 + (2^4 + 13^4 + 7*67^2)/10^4.
a(5797) = 20 with 5797 = 0^4 + (81^4 + 164^4 + 7*4797^2)/20^4.

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, A000583, A214891, A348890, A346643, A347827, A347865, A349942, A349943, A350714, A350857, A350860.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[m=1; Label[bb]; k=m^4; Do[If[SQ[(k*(n-x^4)-y^4-z^4)/7], tab=Append[tab,m]; Goto[aa]],  {x, 0, n^(1/4)}, {y, 0, (k*(n-x^4)/2)^(1/4)},{z,y,(k*(n-x^4)-y^4)^(1/4)}]; m=m+1; Goto[bb]; Label[aa], {n,0,100}];Print[tab]

A351221 Least positive integer m such that m^6*n = x^6 + y^3 + z^2 for some nonnegative integers x,y,z.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 38, 1, 1, 1, 1, 1, 1, 18, 3, 1, 1, 1, 2, 8, 30, 14, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 3, 8, 3, 3, 1, 1, 1, 2, 2, 13, 1, 1, 1, 1, 1, 2, 2, 4, 1, 1, 2, 9, 2, 2, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 1, 11, 9, 2, 3, 1, 1, 1, 1, 1, 3, 3, 1, 26, 1, 2, 2, 1

Offset: 0

Zhi-Wei Sun, Feb 05 2022

nonn

6-3-2 Conjecture: a(n) exists for any nonnegative integer n. Equivalently, each nonnegative rational number can be written as x^6 + y^3 + z^2 with x,y,z nonnegative rational numbers.

			a(6) = 1 with 1^6*6 = 1^6 + 1^3 + 2^2.
a(7) = 38 with 38^6*7 = 42^6 + 1935^3 + 91337^2.
a(21) = 30 with 30^6*21 = 26^6 + 2399^3 + 34545^2.
a(22) = 14 with 14^6*22 = 0^6 + 447^3 + 8737^2.
a(96) = 26 with 26^6*96 = 21^6 + 2711^3 + 98212^2.
a(1120) = 38 with 38^6*1120 = 69^6 + 11499^3 + 1320550^2.
a(2091) = 58 with 58^6*2091 = 161^6 + 39043^3 + 1633994^2.
a(3855) = 51 with 51^6*3855 = 34^6 + 40775^3 + 199008^2.
a(3991) = 45 with 45^6*3991 = 74^6 + 3715^3 + 5738018^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..4000
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, A001014, A350714.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[m=1;Label[bb];k=m^6;Do[If[SQ[k*n-x^6-y^3],tab=Append[tab,m];Goto[aa]],{x,0,(k*n)^(1/6)},{y,0,(k*n-x^6)^(1/3)}];
m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]

cp's OEIS Frontend

A351179 Least positive integer m such that m^6*n = w^6 + x^3 + y^3 + z^3 for some nonnegative integers w,x,y,z.

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A351199 Least positive integer m such that m^3*n = x^3 + y^3 + z^3 for some nonnegative integers x,y,z.

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A351206 Least positive integer m such that n = x^4 + (y^4 + z^4 + 7*w^2)/m^4 for some nonnegative integers x,y,z,w with y <= z.

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A351221 Least positive integer m such that m^6*n = x^6 + y^3 + z^2 for some nonnegative integers x,y,z.

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