A270820 Prime powers p^k (p prime, k > 1) that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

343, 12167, 16807, 29791, 103823, 357911, 493039, 823543, 1092727, 2048383, 3442951, 4657463, 6436343, 6967871, 7880599, 11089567, 13651919, 18191447, 19902511, 28629151, 30080231, 40353607, 46268279, 49430863, 56181887, 80062991, 84604519, 99252847

Offset: 1

Author

Altug Alkan, Mar 23 2016

Keywords

• nonn

Comments

Proper prime powers that are the sum of 4 but no fewer nonzero squares.

This sequence lists the numbers of the form A007522(n)^(2*k+1) where n,k > 0.

Subsequence of A267321.

Terms are 7^3, 23^3, 7^5, 31^3, 47^3, 71^3, 79^3, 7^7, 103^3, 127^3, 151^3, 167^3, 23^5, 191^3, 199^3, ...

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.
		

Crossrefs

Cf. A000961, A004215, A007522, A267321, A246547.

Programs

Mathematica

nn = 120; Select[TakeWhile[Union@ Flatten@ Map[Prime[Range@ nn]^# &, Range[2, Floor[Log2[PrimePi@ nn]^2]]], # <= Prime[nn]^2 &], ! Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]] &] (* Michael De Vlieger, Mar 23 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) ; }
forcomposite(n=4, 1e7, if(isA004215(n) && isprimepower(n), print1(n, ", ")));