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A267321 Perfect powers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

%I A267321 #14 Jan 14 2016 16:27:36
%S A267321 343,3375,12167,16807,21952,29791,59319,103823,166375,216000,250047,
%T A267321 357911,493039,658503,759375,778688,823543,857375,1092727,1367631,
%U A267321 1404928,1685159,1906624,2048383,2460375,2924207,3442951,3796416,4019679,4657463,5359375,6128487
%N A267321 Perfect powers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.
%C A267321 Perfect powers that are the sum of 4 but no fewer nonzero squares. See first comment in A004215.
%C A267321 Intersection of A001597 and A004215.
%C A267321 A134738 is a subsequence.
%C A267321 Motivation for this sequence is the equation m^k = x^2 + y^2 + z^2 where x, y, z are integers and m > 0, k >= 2.
%C A267321 Corresponding exponents are 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, ...
%C A267321 Numbers of the form (4^i*(8*j+7))^(2*k+3) where i,j,k>=0. - _Robert Israel_, Jan 14 2016
%H A267321 Robert Israel, <a href="/A267321/b267321.txt">Table of n, a(n) for n = 1..10000</a>
%e A267321 16807 is a term because 16807 = 7^5 and there is no integer values of x, y and z for the equation 7^5 = x^2 + y^2 + z^2.
%p A267321 N:= 10^10; # to get all terms <= N
%p A267321 sort(convert({seq(seq(seq((4^i*(8*j+7))^(2*k+3),
%p A267321     k=0..floor(1/2*(log[4^i*(8*j+7)](N)-3))),
%p A267321      j = 0 .. floor((N^(1/3)*4^(-i)-7)/8)),
%p A267321 i=0..floor(log[4](N^(1/3)/7)))},list)); # _Robert Israel_, Jan 14 2016
%o A267321 (PARI) isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
%o A267321 for(n=0, 1e7, if(isA004215(n) && ispower(n), print1(n, ", ")));
%Y A267321 Cf. A001597, A004215, A134738.
%K A267321 nonn
%O A267321 1,1
%A A267321 _Altug Alkan_, Jan 13 2016