A304435 Numbers n such that n^5 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).

3, 5, 6, 8, 10, 13, 17, 19, 20, 24, 25, 26, 28, 29, 34, 36, 37, 40, 41, 45, 50, 52, 53, 54, 58, 61, 62, 65, 68, 73, 74, 75, 80, 81, 82, 85, 88, 89, 90, 96, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 130, 136, 137, 145, 146, 148, 149, 150, 153, 157, 160, 164, 168, 169, 170, 173, 176, 178, 180, 181, 185, 192, 193, 194, 197

Offset: 1

Author

M. F. Hasler, May 25 2018

Keywords

• nonn

Comments

Motivated by the search of solutions to a^n + b^(2n+2)/4 = (perfect square), which arises when searching solutions to x^n + y^(n+1) = z^(n+2) of the form x = a*z, y = b*z. It turns out that many solutions are of the form a^n = d (b^(n+1) + d), where d is a perfect power.

Examples

			3^5 = 3^3 + 6^3; 5^5 = 10^2 + 55^2, 6^5 = 2^5 + 88^2, ...
		

Crossrefs

Cf. A304433, A304434, A001597 (perfect powers).

Programs

Maple

LIM:= 200^5: P:={seq(seq(x^k, k=3..floor(log[x](LIM))), x=2..floor(LIM^(1/3)))}:
is_A304435:= proc(n) local n5, Pp;  n5:= n^5;  if remove(t -> subs(t, x)<=1 or subs(t, y)<=1 or subs(t, x-y)=0, [isolve(x^2+y^2=n^5)]) <> [] then return true fi;  Pp:= map(t ->n5-t, P minus {n5, n5/2});  (Pp intersect P <> {}) or (select(issqr, Pp) <> {})
end proc: # adapted from code by Robert Israel for A304434

Mathematica

M = 200; LIM = M^5;
P = Flatten @ Table[Table[x^k, {k, 3, Floor[Log[x, LIM]]}], {x, 2,Floor[ LIM^(1/3)]}];
filterQ[n_] := Module[{n5 = n^5, Pp, x, y}, If[Solve[x > 1 && y > 1 && x != y && x^2 + y^2 == n5, {x, y}, Integers] != {}, Return[True]]; Pp = n5 - (P ~Complement~ {n5, n5/2}); (Pp ~Intersection~ P) != {} || Select[Pp, IntegerQ[Sqrt[#]]&] != {}];
Select[Range[2, M], filterQ] (* Jean-François Alcover, Jun 21 2020, after Maple *)

PARI

N=200;L=N^5;P=List(); for(x=2,sqrtnint(L,3),for(k=3,logint(L,x),listput(P,x^k)));P=Set(P);
is_A304435(n)={for(i=1,#s=sum2sqr(n=n^5),vecmin(s[i])>1 && s[i][1]!=s[i][2] && return(1)); for(i=1,#P, n>P[i]||return; ispower(n-P[i])&& P[i]*2 != n && return(1))} \\ For sum2sqr() see A133388.