A386373 -id:A386373 - cp's OEIS frontend

A386377 a(n) is the number of solutions to the equation x^2 + y^3 + z^4 = w^5 where GCD(x, y, z)=1.

0, 1, 0, 0, 0, 0, 2, 2, 5, 0, 1, 1, 0, 1, 1, 1, 3, 2, 1, 2, 2, 0, 2, 2, 4, 1, 0, 2, 2, 1, 2, 1, 13, 0, 2, 0, 1, 3, 1, 1, 4, 0, 0, 7, 5, 3, 0, 2, 10, 1, 1, 2, 7, 2, 1, 1, 8, 1, 2, 1, 7, 0, 4, 3, 8, 4, 4, 1, 1, 5, 1, 0, 11, 1, 2, 0, 3, 1, 3, 5, 12, 7, 2, 2, 2, 2, 0, 1, 14, 2, 2, 1

Offset: 1

David A. Corneth and Zhining Yang, Jul 20 2025

nonn

			a(9) = 5 because x^2 + y^3 + z^4 = 9^5 where GCD(x,y,z)=1 has 5 positive integer solutions :{220,22,1},{64,38,3},{241,7,5},{9,38,8},{118,29,12}.

Zhining Yang, Table of n, a(n) for n = 1..4500 (terms 1..856 from David A. Corneth)
David A. Corneth, Tuples (x, y, z, w) that are solutions to the equation.
David A. Corneth, PARI program

Cf. A386373, A386521.

f[w_]:=(c=0;zz=w^5;Do[yy=zz-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++]],{y,Floor[yy^(1/3)]}],{z,Floor[zz^(1/4)]}];c);Array[f@#&, 30]

A386521 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 with GCD(x,y,z)=1 has no positive integer solutions.

Original entry on oeis.org

1, 3, 4, 5, 6, 10, 13, 22, 27, 34, 36, 42, 43, 47, 62, 72, 76, 87, 95, 102, 111, 183, 251, 279, 315, 322, 327, 344, 483, 490, 528, 615, 708, 762, 1170, 1302, 2295, 2526, 3282, 3382, 6012

Offset: 1

David A. Corneth and Zhining Yang, Jul 24 2025

a(42) > 6500. - Giovanni Resta, Aug 12 2025

			9 is not a term because 9^5 = x^2 + y^3 + z^4 where GCD(x,y,z)=1 has 5 positive integer solutions: {220,22,1}, {64,38,3}, {241,7,5}, {9,38,8}, {118,29,12}.

Cf. A386373, A386377.

f[w_]:=(c=0;zz=w^5;Do[yy=zz-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++]],{y,Floor[yy^(1/3)]}],{z,Floor[zz^(1/4)]}];c);Select[Range@50,f@#==0&]

a(41) from Giovanni Resta, Aug 12 2025

A387023 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly 5 positive integer solutions.

Original entry on oeis.org

9, 45, 70, 80, 120, 124, 125, 128, 133, 143, 170, 175, 180, 195, 201, 220, 224, 236, 252, 264, 275, 278, 296, 308, 311, 312, 330, 332, 336, 337, 352, 354, 355, 360, 362, 366, 374, 375, 380, 386, 390, 394, 399, 404, 411, 416, 418, 428, 430, 444, 461, 466, 477, 484, 488, 500

Offset: 1

Zhining Yang, Aug 13 2025

			444 is in the sequence because 444^5 = x^2 + y^3 + z^4 where GCD (x, y, z) = 1 has exactly 5 positive integer solutions: {676786, 25603, 343}, {342332, 25775, 345}, {4123199, 5503, 544}, {2451712, 21919, 919}, {3889117, 679, 1208}.

Zhining Yang, Table of n, a(n) for n = 1..328

Cf. A386373, A386377, A386521.

Do[w5=w^5;s={};c=0;
Do[yy=w5-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++;AppendTo[s,{x,y,z}]]],{y,Floor[yy^(1/3)]}],{z,Floor[w5^(1/4)]}];
If[c==5,Print[w,s]],{w,100}]

A386988 a(n) is the smallest integer w such that the equation x^2 + y^4 + z^6 = w^8 where GCD(x,y,z)=1 has exactly n positive integer solutions.

Original entry on oeis.org

25, 9, 17, 53, 3

Offset: 1

Zhining Yang, Aug 12 2025

a(6)>1024.

			a(3) = 17 because 17^8 = 36840^2 + 273^4 + 20^6 = 82367^2 + 24^4 + 24^6 = 48^2 + 287^4  + 24^6 and for no integer smaller than 17 we have 3 solutions.

Fang, How to find non-trivial positive integer solutions to the Diophantine equation a^2 + b^4 + c^6 = d^8, where gcd(a,b,c) = 1?.

Cf. A386373.

f[w_]:=(v={};c=0;w8=w^8;
Do[yy=w8-z^6;Do[xx=yy-y^4;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,AppendTo[v,{x,y,z}];c++]],{y,Floor[yy^(1/4)]}],{z,Floor[w8^(1/6)]}];{c,w,v});
s=Table[{},5];
For[k=1,k<=60,k++,r=f[k][[1]];If[s[[r]]=={},s[[r]]=f[k];Print[s[[r]]]]]

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A386377 a(n) is the number of solutions to the equation x^2 + y^3 + z^4 = w^5 where GCD(x, y, z)=1.

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A386521 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 with GCD(x,y,z)=1 has no positive integer solutions.

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A387023 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly 5 positive integer solutions.

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Mathematica

A386988 a(n) is the smallest integer w such that the equation x^2 + y^4 + z^6 = w^8 where GCD(x,y,z)=1 has exactly n positive integer solutions.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica