%I #21 Aug 16 2020 09:58:09

%S 30,85,195,240,525,700,960,1020,1170,1845,2720,3045,3120,4026,4818,

%T 5005,5655,6240,7290,7680,7735,7770,9717,11349,11570,12432,13560,

%U 15015,16240,16695,16800,17476,20655,22185,22400,22605,23730,27768,30720,32120,32640,32760,34390,37440,40885,42105,43095

%N Numbers n such that the Diophantine equation x^4+y^5=n^4 has solutions.

%C From _Robert Israel_, Oct 29 2019: (Start)

%C The first number that occurs in more than one way is 1845, as 205^4 + 410^5 = 1476^4 + 369^5 = 1845^4.

%C If n is a term then so is x^5*n for any x.

%C Includes s*(s^4-t^4)/u^5 if s>t>0 and s^4 == t^4 (mod u^5), as x=t*(s^4-t^4)/u^5, y=(s^4-t^4)/u^4, n=s*(s^4-t^4)/u^5 is a solution.

%C (End)

%H Robert Israel, <a href="/A070756/b070756.txt">Table of n, a(n) for n = 1..100</a>

%p filter:= proc(n) local n4, y,s;

%p n4:= n^4;

%p for y from ceil((n4-(n-1)^4)^(1/5)) do

%p s:= n4-y^5;

%p if s <= 0 then return false fi;

%p if issqr(s) and issqr(sqrt(s)) then return true fi

%p od

%p end proc:

%p select(filter, [$1..10000]); # _Robert Israel_, Oct 29 2019

%t filterQ[n_] := Module[{n4 = n^4, y, s}, For[y = Ceiling[(n4 - (n-1)^4 )^(1/5)], True, y++, s = n4 - y^5; If[s <= 0, Return[False]]; If[ IntegerQ[Sqrt[s]] && IntegerQ[Sqrt[Sqrt[s]]], Return[True]]]];

%t Reap[Do[If[filterQ[n], Print[n]; Sow[n]], {n, 10000}]][[2, 1]] (* _Jean-François Alcover_, Aug 16 2020, after Maple *)

%o (PARI) for(n=0,500,if(sum(i=1,n,sum(j=1,i,if(i^4+j^5-n^4,0,1)))>0,print1(n,",")))

%K nonn

%O 1,1

%A _Benoit Cloitre_, May 14 2002

%E More terms from _John W. Layman_, Nov 25 2002

%E More terms from _Robert Israel_, Oct 29 2019