A386521 -id:A386521 - cp's OEIS frontend

A386373 a(n) is the smallest integer w such that the equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly n positive integer solutions.

2, 7, 17, 25, 9, 100, 44, 57, 117, 49, 73, 81, 33, 89, 177, 193, 305, 161, 257, 273, 425, 289, 697, 441, 313, 689, 369, 593, 809, 233, 761, 1865, 2001, 857, 1121, 649, 1353, 865, 521, 1257, 577, 681, 2081, 1409, 1169, 1753, 1801, 1201, 1745, 2833, 3853, 3649, 3353, 1305, 793

Offset: 1

Zhining Yang, Jul 19 2025

nonn

From David A. Corneth, Jul 20 2025: (Start)
a(41) = 577. If a(41) is 1 (mod 8) then that values is exact.
For 10 <= n <= 30 we have a(n) == 1 (mod 8).
Heuristically this is no coincidence. There are 8^3 = 512 tuples (x, y, z) mod 8. The frequencies of k (mod 8) for x^2 + y^3 + z^4 for k = 0 through 7 are 64, 128, 96, 32, 64, 64, 32, 32 respectively. So 1 (mod 8) has the single largest value at 128 such tuples.
Extending this to other moduli like 56 we get the largest frequencies (7168) come from 9, 17, 25 and 33 (mod 56).
The second largest frequency is 6272 which occurs at 49 (mod 56). For n = 3, 4 and 10 <= n <= 20, 22, 30 we have a(n) == 9, 17, 25, 33 or 49 (mod 56). (End)

			a(4) = 25 because 25^5 = 1852^2 + 185^3 + 8^4 = 2711^2 + 134^3 + 10^4 = 2472^2 + 150^3 + 23^4 = 2973^2 + 15^3 + 31^4 and no integer less than 25 has 4 solutions.

Zhining Yang, Table of n, a(n) for n = 1..70

Cf. A386377, A386521.

f[w_]:=(v={};c=0;nn=w^5;
Do[yy=nn-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,AppendTo[v,{x,y,z,d}];c++]],{y,Floor[yy^(1/3)]}],{z,Floor[nn^(1/4)]}];{c,w,v});
s=Table[{},20];
For[k=1,k<=100,k++,r=f[k][[1]];If[s[[r]]=={},s[[r]]=g[k];Print[s[[r]]]]]

a(21)-a(31) from David A. Corneth, Jul 20 2025

a(32)-a(58) from Zhining Yang, Jul 31 2025

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.

Original entry on oeis.org

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]

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}]

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A386373 a(n) is the smallest integer w such that the equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly n positive integer solutions.

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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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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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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica