A390055 Powers k^m, m > 0, of nonsquarefree numbers k that are not squareful.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198

Offset

1,1

Comments

Union of A386762 and A332785, where the former contains the perfect powers of the latter.

Analogous to A182853, union of A303606 and A120944.

A059404 is the union of this sequence and A389959.

A024619 is the union of this sequence, A389959, and A182853.

A013929 is the union of this sequence, A389959, and A072777.

Proper subset of A126706.

Starts to differ from A317616 at a(4071). - R. J. Mathar, Nov 02 2025

Links

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

Example

Table of n, a(n) for select n:
  n   a(n)
----------------------------
  1    12 = 2^2 * 3
  2    18 = 2   * 3^2
  3    20 = 2^2 * 5
  4    24 = 2^3 * 3
  5    28 = 2^2 * 7
  6    40 = 2^3 * 5
  7    44 = 2^2 * 11
  8    45 = 3^2 * 5
  9    48 = 2^4 * 3
 38   144 = 2^4 * 3^2 = 12^2
 99   324 = 2^2 * 3^4 = 18^2
128   400 = 2^4 * 5^2 = 20^2

Mathematica

nn = 200; i = k = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range[nn], 1 == Min[#] < Max[#] &@ FactorInteger[#][[All, -1]] &] ]; Union@ Reap[While[j = 1; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 1, k++; i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot
from oeis_sequences.OEISsequences import bisection, squarefreepi
def A390055(n):
    def g(x):
        c, l, j = 1+x-squarefreepi(integer_nthroot(x, 3)[0])-squarefreepi(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-g(x)-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f, n, n) # Chai Wah Wu, Oct 31 2025

Crossrefs

Cf. A013929, A024619, A059404, A072777, A126706, A182853, A332785, A386762, A389959.

Sequence in context: A242416 A360248 A317616 * A375934 A332785 A393946

Adjacent sequences: A390052 A390053 A390054 * A390056 A390057 A390058

Keyword

nonn

Author

Michael De Vlieger, Oct 24 2025

Status

approved