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%I A070756 #21 Aug 16 2020 09:58:09
%S A070756 30,85,195,240,525,700,960,1020,1170,1845,2720,3045,3120,4026,4818,
%T A070756 5005,5655,6240,7290,7680,7735,7770,9717,11349,11570,12432,13560,
%U A070756 15015,16240,16695,16800,17476,20655,22185,22400,22605,23730,27768,30720,32120,32640,32760,34390,37440,40885,42105,43095
%N A070756 Numbers n such that the Diophantine equation x^4+y^5=n^4 has solutions.
%C A070756 From _Robert Israel_, Oct 29 2019: (Start)
%C A070756 The first number that occurs in more than one way is 1845, as 205^4 + 410^5 =  1476^4 + 369^5 = 1845^4.
%C A070756 If n is a term then so is x^5*n for any x.
%C A070756 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 A070756 (End)
%H A070756 Robert Israel, <a href="/A070756/b070756.txt">Table of n, a(n) for n = 1..100</a>
%p A070756 filter:= proc(n) local n4, y,s;
%p A070756   n4:= n^4;
%p A070756   for y from ceil((n4-(n-1)^4)^(1/5)) do
%p A070756     s:= n4-y^5;
%p A070756     if s <= 0 then return false fi;
%p A070756     if issqr(s) and issqr(sqrt(s)) then return true fi
%p A070756   od
%p A070756 end proc:
%p A070756 select(filter, [$1..10000]); # _Robert Israel_, Oct 29 2019
%t A070756 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 A070756 Reap[Do[If[filterQ[n], Print[n]; Sow[n]], {n, 10000}]][[2, 1]] (* _Jean-François Alcover_, Aug 16 2020, after Maple *)
%o A070756 (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 A070756 nonn
%O A070756 1,1
%A A070756 _Benoit Cloitre_, May 14 2002
%E A070756 More terms from _John W. Layman_, Nov 25 2002
%E A070756 More terms from _Robert Israel_, Oct 29 2019

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