A304436 - OEIS

%I #11 Aug 12 2020 11:27:22

%S 5,10,13,15,17,20,25,26,29,30,33,34,35,36,37,39,40,41,45,50,51,52,53,

%T 55,56,58,60,61,65,68,70,73,74,75,78,80,81,82,85,87,89,90,91,95,96,97,

%U 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

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

%C 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.

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

%p LIM:=200^6: P:={seq(seq(x^k, k=3..floor(log[x](LIM))), x=2..floor(LIM^(1/3)))}:

%p 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) <> {})

%p end proc: # adapted from code by _Robert Israel_ for A304434

%t LIM = 200^6;

%t P = Union@ Flatten@ Table[Table[x^k, {k, 3, Floor[Log[x, LIM]]}], {x, 2, Floor[LIM^(1/3)]}];

%t 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[#]]&] != {}];

%t Reap[For[n = 1, n <= 150, n++, If[filterQ[n], Print[n]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Aug 12 2020, after Maple *)

%o (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.

%o 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.

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

%K nonn

%O 1,1

%A _M. F. Hasler_, May 25 2018