OFFSET
1,1
COMMENTS
From Robert Israel, Oct 29 2019: (Start)
The first number that occurs in more than one way is 1845, as 205^4 + 410^5 = 1476^4 + 369^5 = 1845^4.
If n is a term then so is x^5*n for any x.
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.
(End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..100
MAPLE
filter:= proc(n) local n4, y, s;
n4:= n^4;
for y from ceil((n4-(n-1)^4)^(1/5)) do
s:= n4-y^5;
if s <= 0 then return false fi;
if issqr(s) and issqr(sqrt(s)) then return true fi
od
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 29 2019
MATHEMATICA
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]]]];
Reap[Do[If[filterQ[n], Print[n]; Sow[n]], {n, 10000}]][[2, 1]] (* Jean-François Alcover, Aug 16 2020, after Maple *)
PROG
(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, ", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, May 14 2002
EXTENSIONS
More terms from John W. Layman, Nov 25 2002
More terms from Robert Israel, Oct 29 2019
STATUS
approved