Diophantine equation x+y^2+z^3=n^4

[Max A.]

Zak,

First 50 terms of your sequence are:

0, 5, 27, 75, 163, 303, 521, 815, 1223, 1753, 2419, 3245, 4246, 5445,
6857, 8541, 10459, 12675, 15187, 18049, 21230, 24829, 28820, 33232,
38092, 43417, 49243, 55634, 62573, 70096, 78220, 86963, 96359, 106501,
117325, 128873, 141269, 154403, 168434, 183255, 199048, 215654,
233339, 251984, 271589, 292240, 314034, 336915, 360930, 386106

Max

On 9/9/06, zak seidov <zakseidov at yahoo.com> wrote:
> Dear seqfans,
>
> I've just submitted:
>
> solutions to equation x+y^2+z^3=n^4 (see below).
>
> 1) Can anyone check/extend entries?
>
> 2) What about case of all distinct x,y,z:
>    (x-y)(x-z)(y-z)=/=0?
>
> 2a) n=2 3 (of total 5) solns:
>     {x,y,z}={6,3,1},{7,1,2},{11,2,1};
>
> 2b) n=2 23 (of total 27) solns:
> {x,y,z}={5,7,3},{8,3,4},{9,8,2},{13,2,4},{16,1,4},{16,8,1},{18,6,3},{24,7,2},{29,5,3},{31,7,1},{37,6,2},{38,4,3},{44,6,1},{48,5,2},{50,2,3},{53,1,3},{55,5,1},{57,4,2},{64,3,2},{64,4,1},{71,3,1},{72,1,2},{76,2,1}
>
> 2c) n=1..10: 0,3,23,69,155,293,508,799,1205,1732 solns
>
>
> thanks, Zak
>
>
> %I A000001
> %S A000001 0, 5, 27, 75, 163, 303, 521, 815, 1223,
> 1753
> %N A000001 a(n) = number of solutions to the
> Diophantine equation
> x+y^2+z^3=n^4 with positive x,y,z.
> %e A000001 a(2)=5 because there are five solutions to
> equation
> x+y^2+z^3=2^4 with
> {x,y,z}={4,2,2},{6,3,1},{7,1,2},{11,2,1},{14,1,1};
> a(3)=27 because there are 27 solutions to equation
> x+y^2+z^3=3^4 with
> {x,y,z}={1,4,4},{5,7,3},{8,3,4},{9,8,2},{13,2,4},{16,1,4},{16,8,1},{18,6,3},{24,7,2},{29,5,3},{31,7,1},{37,6,2},{38,4,3},{44,6,1},{45,3,3},{48,5,2},{50,2,3},{53,1,3},{55,5,1},{57,4,2},{64,3,2},{64,4,1},{69,2,2},{71,3,1},{72,1,2},{76,2,1},{79,1,1}
> %O A000001 1
> %K A000001 ,more,nice,nonn,
> %A A000001 Zak Seidov (zakseidov at yahoo.com), Sep 09
> 2006
>
>
>
> __________________________________________________
> Do You Yahoo!?
> Tired of spam?  Yahoo! Mail has the best spam protection around
> http://mail.yahoo.com
>