A304434 Numbers n such that n^4 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).

3, 5, 6, 9, 10, 12, 13, 14, 15, 17, 20, 24, 25, 26, 28, 29, 30, 34, 35, 36, 37, 39, 40, 41, 42, 45, 48, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 65, 68, 70, 71, 72, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 96, 97, 98, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113

Offset: 1

Author

M. F. Hasler, May 22 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

			3^4 = 2^5 + 7^2; 5^4 = 7^2 + 24^2, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1301

Crossrefs

Cf. A304433, A001597 (perfect powers), A076467 (third or higher powers).

Programs

Maple

N:= 200: # to get terms <= N
N4:= N^4:
P:= {seq(seq(x^k,k=3..floor(log[x](N4))),x=2..floor(N4^(1/3)))}:
filter:= proc(n) local n4, Pp;
  n4:= n^4;
  if remove(t -> subs(t,x)<=1 or subs(t,y)<=1 or subs(t,x-y)=0, [isolve(x^2+y^2=n4)]) <> [] then return true fi;
  Pp:= map(t ->n4-t, P minus {n4, n4/2});
  (Pp intersect P <> {}) or (select(issqr,Pp) <> {})
end proc:
A:= select(filter, [$2..N]); # Robert Israel, May 24 2018

PARI

L=200^4; 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_A304434(n)={for(i=1, #s=sum2sqr(n=n^4), 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))} \\ The above P must be computed up to L >= n^4. For sum2sqr() see A133388.