A267321 -id:A267321 - cp's OEIS frontend

A267986 Perfect powers of the form x^2 + y^2 + z^2 where x > y > z > 0.

49, 81, 121, 125, 169, 196, 216, 225, 243, 289, 324, 361, 441, 484, 529, 625, 676, 729, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2187, 2197, 2209, 2401, 2500, 2601, 2704, 2744, 2809, 2916, 3025, 3125, 3136

Offset: 1

Altug Alkan, Jan 23 2016

nonn

Intersection of A001597 and A004432.
Note that this sequence is not the complement of A267321. This sequence is a subsequence for complement of A267321.
Sequence focuses on the equation m^k = x^2 + y^2 + z^2 where x > y > z > 0 and m > 0, k >= 2.
Corresponding exponents are 2, 4, 2, 3, 2, 2, 3, 2, 5, 2, 2, 2, 2, 2, 2, 4, 2, 6, 2, 2, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 3, 2, 4, 2, 2, ...

			49 is a term because 49 = 7^2 = 2^2 + 3^2 + 6^2.
81 is a term because 81 = 9^2 = 1^2 + 4^2 + 8^2.
121 is a term because 121 = 11^2 = 2^2 + 6^2 + 9^2.

Cf. A001597, A004432, A266927, A267321.

fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Range@ 1800, fQ@ # && Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]] &] (* Michael De Vlieger, Jan 24 2016, after Ant King at A001597 *)

isA004432(n) = for(x=1, sqrtint(n\3), for(y=x+1, sqrtint((n-1-x^2)\2), issquare(n-x^2-y^2) && return(1)));
for(n=1, 1e4, if(isA004432(n) && ispower(n), print1(n, ", ")));

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.

Original entry on oeis.org

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

Altug Alkan, Mar 23 2016

nonn

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, ...

			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.

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

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 *)

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, ", ")));

cp's OEIS Frontend

A267986 Perfect powers of the form x^2 + y^2 + z^2 where x > y > z > 0.

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