A266918 Perfect power Löschian numbers.

1, 4, 9, 16, 25, 27, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025, 2116, 2187, 2197, 2209, 2304, 2401, 2500

Offset

1,2

Comments

Inspired by A266836. See first comment in A266836.

Intersection of A001597 and A003136.

Obviously, this sequence contains all positive squares.

Perfect powers that are not the Löschian numbers are 8, 32, 125, 128, 216, 512, 1000, 1331, 2048, 2744, 3125, 3375, 4913, 5832, 7776, ...

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

25 is a term because 25 = 5^2 = 5^2 + 5*0 + 0^2.
27 is a term because 27 = 3^3 = 3^2 + 3*3 + 3^2.
243 is a term because 243 = 3^5 = 9^2 + 9*9 + 9^2.
343 is a term because 343 = 7^3 = 18^2 + 18*1 + 1^2.

Mathematica

fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; gQ[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x y + y^2, {x, y}, Integers]]]; Select[Range@ 2500, fQ@# && gQ@# &] (* Michael De Vlieger, Jan 06 2016, after Ant King at A001597 and Jean-François Alcover at A003136 *)

Prog

(PARI) x='x+O('x^10^4); p=eta(x)^3/eta(x^3); for(n=0, 9999, if(polcoeff(p, n) != 0 && (ispower(n) || n==1), print1(n, ", ")));
(PARI) is(n) = (ispower(n) || n==1) && #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=0, 1e4, if(is(n), print1(n, ", ")));

Crossrefs

Cf. Loeschian numbers: A003136 (all), A266836 (2*k+1), A202822 (3*k+1), A260682 (6*k+1).

Cf. A001597.

Sequence in context: A061077 A292675 A254719 * A086132 A010433 A175592

Adjacent sequences: A266915 A266916 A266917 * A266919 A266920 A266921

Keyword

nonn

Author

Altug Alkan, Jan 06 2016

Status

approved