A304436 Numbers n such that n^6 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 33, 34, 35, 36, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 56, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 81, 82, 85, 87, 89, 90, 91, 95, 96, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 112, 113, 115, 116, 117, 119, 120, 122, 123, 125, 126, 130, 135, 136, 137, 140, 143, 145, 146, 148, 149, 150

Offset: 1

Author

M. F. Hasler, May 25 2018

Keywords

• nonn

Comments

Motivated by the search for solutions to a^n + b^(2n+2)/4 = (perfect square), which arises when searching for 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

			5^6 = 35^2 + 120^2, 10^6 = 280^2 + 960^2, ...
		

Crossrefs

Cf. A304433, A304434, A304435, A001597 (perfect powers).

Programs

Maple

LIM:=200^6: P:={seq(seq(x^k, k=3..floor(log[x](LIM))), x=2..floor(LIM^(1/3)))}:
is_A304436:= proc(n) local N, S;  N:= n^6;  if remove(t -> subs(t, x)<=1 or subs(t, y)<=1 or subs(t, x-y)=0, [isolve(x^2+y^2=N)]) <> [] then return true fi;  S:= map(t ->N-t, P minus {N, N/2});  (S intersect P <> {}) or (select(issqr, S) <> {})
end proc: # adapted from code by Robert Israel for A304434

Mathematica

LIM = 200^6;
P = Union@ Flatten@ Table[Table[x^k, {k, 3, Floor[Log[x, LIM]]}], {x, 2, Floor[LIM^(1/3)]}];
filterQ[n_] := Module[{M = n^6, S}, If[Solve[x > 1 && y > 1 && x^2 + y^2 == M, {x, y}, Integers] != {}, Return [True]]; S = M - (P ~Complement~ {M, M/2}); S ~Intersection~ P != {} || Select[S, IntegerQ[Sqrt[#]]&] != {}];
Reap[For[n = 1, n <= 150, n++, If[filterQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Aug 12 2020, after Maple *)

PARI

L=200^6;P=List(); for(x=2,sqrtnint(L,3),for(k=3,logint(L,x),listput(P,x^k)));#P=Set(P) \\ This P = A076467 \ {1} = A111231 \ {0} up to limit L.
is(n,e=6)={for(i=1,#s=sum2sqr(n=n^e),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))} \\ Needs the above P computed up to L >= n^6. For sum2sqr() see A133388.