A266927 Perfect powers of the form x^2 + y^2 where x and y are positive integers.

8, 25, 32, 100, 125, 128, 169, 225, 289, 400, 512, 625, 676, 841, 900, 1000, 1156, 1225, 1369, 1521, 1600, 1681, 2025, 2048, 2197, 2500, 2601, 2704, 2809, 3025, 3125, 3364, 3600, 3721, 4225, 4624, 4900, 4913, 5329, 5476, 5625, 5832, 6084, 6400, 6724, 7225, 7569

Offset: 1

Author

Altug Alkan, Jan 06 2016

Keywords

• nonn

Comments

Intersection of A000404 and A001597.

A134422 is a subsequence.

Obviously, this sequence contains all numbers of the form 2^(2*n+1), for n > 0.

Motivation for this sequence is the equation m^k = x^2 + y^2 where m,x,y > 0, k >= 2. - Altug Alkan, Jan 11 2016

Examples

			25 is a term because 25 = 5^2 = 3^2 + 4^2.
32 is a term because 32 = 2^5 = 4^2 + 4^2.
125 is a term because 125 = 5^3 = 10^2 + 5^2.
169 is a term because 169 = 13^2 = 5^2 + 12^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000404, A001597, A004171, A134422.

Programs

Maple

N:= 10000: # to get all terms <= N
g:= proc(k)
    local F,F1,F2,F3,f;
    F:= ifactors(k)[2];
    F2,F:= selectremove(f->f[1]=2,F);
    F1,F3:= selectremove(f -> f[1] mod 4 = 1, F);
    if F1 <> [] then
       if hastype(map(f -> f[2],F3),odd) then
          seq(k^j, j=2..floor(log[k](N)),2)
       else seq(k^j, j=2..floor(log[k](N)))
       fi
    elif F2 = [] or F2[1][2]::even or hastype(map(f -> f[2],F3),odd) then NULL
    else seq(k^j, j=3..floor(log[k](N)),2)
    fi
end proc:
sort(convert(map(g,{$2..floor(sqrt(N))}),list)); # Robert Israel, Jan 11 2016

Mathematica

lim = 7600; fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Union@ Flatten@ Table[a^2 + b^2, {a, Floor[Sqrt[lim - 1]]}, {b, a, Floor[Sqrt[lim - a^2]]}], fQ] (* Michael De Vlieger, Jan 06 2016, after N. J. A. Sloane and J. H. Conway at A000404 and Ant King at A001597 *)

PARI

is(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
for(n=1, 1e4, if((ispower(n) || n==1) && is(n), print1(n, ", ")));