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

1,1

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

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

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

Cf. A001597, A004215, A134738.

Sequence in context: A250138 A016923 A016983 * A134738 A017151 A017247

Adjacent sequences: A267318 A267319 A267320 * A267322 A267323 A267324

Keyword

nonn

Author

Altug Alkan, Jan 13 2016

Status

approved