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