login
A266918
Perfect power Löschian numbers.
1
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
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
KEYWORD
nonn
AUTHOR
Altug Alkan, Jan 06 2016
STATUS
approved