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

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

Author

Altug Alkan, Jan 13 2016

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A001597, A004215, A134738.

Programs

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

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, ", ")));