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

%I A266927 #25 Jan 11 2016 21:38:09
%S A266927 8,25,32,100,125,128,169,225,289,400,512,625,676,841,900,1000,1156,
%T A266927 1225,1369,1521,1600,1681,2025,2048,2197,2500,2601,2704,2809,3025,
%U A266927 3125,3364,3600,3721,4225,4624,4900,4913,5329,5476,5625,5832,6084,6400,6724,7225,7569
%N A266927 Perfect powers of the form x^2 + y^2 where x and y are positive integers.
%C A266927 Intersection of A000404 and A001597.
%C A266927 A134422 is a subsequence.
%C A266927 Obviously, this sequence contains all numbers of the form 2^(2*n+1), for n > 0.
%C A266927 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
%H A266927 Robert Israel, <a href="/A266927/b266927.txt">Table of n, a(n) for n = 1..10000</a>
%e A266927 25 is a term because 25 = 5^2 = 3^2 + 4^2.
%e A266927 32 is a term because 32 = 2^5 = 4^2 + 4^2.
%e A266927 125 is a term because 125 = 5^3 = 10^2 + 5^2.
%e A266927 169 is a term because 169 = 13^2 = 5^2 + 12^2.
%p A266927 N:= 10000: # to get all terms <= N
%p A266927 g:= proc(k)
%p A266927     local F,F1,F2,F3,f;
%p A266927     F:= ifactors(k)[2];
%p A266927     F2,F:= selectremove(f->f[1]=2,F);
%p A266927     F1,F3:= selectremove(f -> f[1] mod 4 = 1, F);
%p A266927     if F1 <> [] then
%p A266927        if hastype(map(f -> f[2],F3),odd) then
%p A266927           seq(k^j, j=2..floor(log[k](N)),2)
%p A266927        else seq(k^j, j=2..floor(log[k](N)))
%p A266927        fi
%p A266927     elif F2 = [] or F2[1][2]::even or hastype(map(f -> f[2],F3),odd) then NULL
%p A266927     else seq(k^j, j=3..floor(log[k](N)),2)
%p A266927     fi
%p A266927 end proc:
%p A266927 sort(convert(map(g,{$2..floor(sqrt(N))}),list)); # _Robert Israel_, Jan 11 2016
%t A266927 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 *)
%o A266927 (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))}
%o A266927 for(n=1, 1e4, if((ispower(n) || n==1) && is(n), print1(n, ", ")));
%Y A266927 Cf. A000404, A001597, A004171, A134422.
%K A266927 nonn
%O A266927 1,1
%A A266927 _Altug Alkan_, Jan 06 2016