A304432 Numbers n such that n^2 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).

5, 6, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 24, 25, 26, 28, 29, 30, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 48, 50, 51, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 68, 70, 71, 72, 73, 74, 75, 76, 78, 80, 81, 82, 85, 87, 89, 90, 91, 95, 96, 97, 98, 99, 100

Offset: 1

Author

M. F. Hasler, May 13 2018

Keywords

• nonn

Examples

			5^2 = 25 = 3^2 + 4^2; 6^2 = 3^2 + 3^3; 9^2 = 2^5 + 7^2, ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597 (perfect powers), A282550.

Programs

Mathematica

n = 120; i = 1;
s = Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  GCD @@ FactorInteger[#][[All, -1]] > 1 &];
m = Length[s];
Union@ Reap[
  While[i <= m, j = i + 1;
    While[k = s[[i]] + s[[j]]; k <= nn,
      If[And[IntegerQ@ Sqrt[k], i != j],
      Sow[Sqrt[k]]]; j++];
i++] ][[-1, 1]] (* Michael De Vlieger, Dec 02 2024 *)

PARI

is(n)=for(i=2,(n^2-1)\2, ispower(i)&&ispower(n^2-i)&&return(i)) \\ For more efficiency, loop over elements of precomputed A001597\{1}.

PARI

L=100; PP=List(); a=Set(); for(n=1,L^2, ispower(n)||next; for(i=1,#PP, issquare(n+PP[i],&m)&& m<=L&&  a=setunion(a,[m])); listput(PP,n)); a