A389864 Powers k^m, m > 1, such that k is neither squarefree nor perfect power.

144, 324, 400, 576, 784, 1600, 1728, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 5832, 6400, 7056, 7744, 8000, 8100, 8464, 9216, 9604, 9801, 10816, 11664, 12544, 13456, 13689, 13824, 14400, 15376, 15876, 17424, 18225, 18496, 19600

Offset

1,1

Comments

Subset of A131605, in turn a subset of A286708, a subset of A126706.

Union of A383394 and A386762 = A001597 \ A072777 = A059404 \ A303946 = A131605 \ A303606.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Example

Table of n, a(n) for select n:
 n     a(n)
--------------------------------------
 1     144 =  12^2 = 2^4 *  3^2
 2     324 =  18^2 = 2^2 *  3^4
 3     400 =  20^2 = 2^4 *  5^2
 4     576 =  24^2 = 2^6 *  3^2
 5     784 =  28^2 = 2^4 *  7^2
 6    1600 =  40^2 = 2^6 *  5^2
 7    1728 =  12^3 = 2^6 *  3^3
 8    1936 =  44^2 = 2^4 * 11^2
 9    2025 =  45^2 = 3^4 *  5^2
15    3600 =  60^2 = 2^4 *  3^2 * 5^2
18    5184 =  72^2 = 2^6 *  3^4
32   11664 = 108^2 = 2^4 *  3^6

Mathematica

nn = 20000; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[Max[#] > 1, GCD @@ # == 1] &@ FactorInteger[x][[;; , -1]]; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], fQ] ]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++] ][[-1, 1]]

Prog

(PARI) isok(n) = if (ispower(n, , &np), !issquarefree(np) && !ispower(np)); \\ Michel Marcus, Oct 23 2025
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A389864(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return int(n+x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1)+g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length())))
    return bisection(f, n, n) # Chai Wah Wu, Oct 23 2025

Crossrefs

Cf. A001597, A001694, A013929, A059404, A072777, A131605, A303606, A303946, A383394, A386762.

Sequence in context: A335543 A366854 A380857 * A386762 A390379 A392424

Adjacent sequences: A389861 A389862 A389863 * A389865 A389866 A389867

Keyword

nonn

Author

Michael De Vlieger, Oct 23 2025

Status

approved