A390823 Powers k^m, m > 1, of numbers k that are neither prime powers nor powerful.

36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624, 4761, 4900, 5476

Offset

1,1

Comments

Perfect powers of numbers in A389065, i.e., composite weak numbers.

Numbers that are not prime powers and whose prime exponents are not coprime.

A131605(48) = A383394(1) = 5184 is the first term not in this sequence.

Union of A386762 and A303606.

A001597 is the disjoint union of {1}, this sequence, A383394, and A246547.

A001694 is the disjoint union of {1}, this sequence, A389959, and A246547.

A131605 is the disjoint union of {1}, this sequence, and A383394.

A286708 is the disjoint union of this sequence and A389959.

A126706 is the disjoint union of this sequence, A383394, and A303946.

This sequence is the disjoint union of A387254 (squares in this sequence) and A390905.

Links

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

Index entries for sequences related to powerful numbers.

Example

Let s = A303606 and let t = A386762.
Table of n, a(n) for select n:
 n   a(n)
-----------------------------------------
 1    36 =  6^2 = s(1) = 2^2 *  3^2
 2   100 = 10^2 = s(2) = 2^2 *  5^2
 3   144 = 12^2 = t(1) = 2^4 *  3^2
 4   196 = 14^2 = s(3) = 2^2 *  7^2
 5   216 =  6^3 = s(4) = 2^3 *  3^3
 6   225 = 15^2 = s(5) = 3^2 *  5^2
 7   324 = 18^2 = t(2) = 2^2 *  3^4
 8   400 = 20^2 = t(3) = 2^4 *  5^2
 9   441 = 21^2 = s(6) = 3^2 *  7^2
10   484 = 22^2 = s(7) = 2^2 * 11^2
11   576 = 24^2 = t(4) = 2^6 *  3^2
14   900 = 30^2 = s(9) = 2^2 *  3^2 * 5^2

Mathematica

Select[Range[5000], And[Length[#] > 1, GCD @@ # > 1] &[FactorInteger[#][[;; , -1]] ] &] (* or *)
nn = 5000; mm = Sqrt[nn]; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Complement[Select[Range[mm], Not @* PrimePowerQ], Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]} ] ] ]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot, primepi
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A390823(n):
    def g(x):
        c, l, j = x-squarefreepi(integer_nthroot(x, 3)[0])-primepi(x), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(l-w)
            l, j = w, isqrt(x//k2**3)
        return c+l
    def f(x): return n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f, n+1, n+1) # Chai Wah Wu, Dec 02 2025

Crossrefs

Cf. A001597, A001694, A024619, A120944, A126706, A246547, A286708, A303606, A332785, A383394, A386762, A387254, A388304, A388549, A389065, A389959, A390905.

Sequence in context: A072413 A377817 A131605 * A391756 A376119 A387254

Adjacent sequences: A390820 A390821 A390822 * A390824 A390825 A390826

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 23 2025

Status

approved