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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.
3
343, 3375, 12167, 16807, 21952, 29791, 59319, 103823, 166375, 216000, 250047, 357911, 493039, 658503, 759375, 778688, 823543, 857375, 1092727, 1367631, 1404928, 1685159, 1906624, 2048383, 2460375, 2924207, 3442951, 3796416, 4019679, 4657463, 5359375, 6128487
OFFSET
1,1
COMMENTS
Perfect powers that are the sum of 4 but no fewer nonzero squares. See first comment in A004215.
Intersection of A001597 and A004215.
A134738 is a subsequence.
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.
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, ...
Numbers of the form (4^i*(8*j+7))^(2*k+3) where i,j,k>=0. - Robert Israel, Jan 14 2016
LINKS
EXAMPLE
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.
MAPLE
N:= 10^10; # to get all terms <= N
sort(convert({seq(seq(seq((4^i*(8*j+7))^(2*k+3),
k=0..floor(1/2*(log[4^i*(8*j+7)](N)-3))),
j = 0 .. floor((N^(1/3)*4^(-i)-7)/8)),
i=0..floor(log[4](N^(1/3)/7)))}, list)); # Robert Israel, Jan 14 2016
PROG
(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, ", ") ; ) ; ) ; }
for(n=0, 1e7, if(isA004215(n) && ispower(n), print1(n, ", ")));
CROSSREFS
KEYWORD
nonn
AUTHOR
Altug Alkan, Jan 13 2016
STATUS
approved