A389065 Numbers that are neither prime powers nor powerful.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 106

Offset

1,1

Comments

Composite weak numbers, where weak signifies numbers that are not powerful.

Proper subset of A106543; 72 is in A106543, but since 72 is powerful, it is not in this sequence.

Links

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

Formula

Union of A120944 (squarefree composites) and A332785 (nonsquarefree numbers that are not powerful).

Intersection of A002808 and A052485.

This sequence is A024619 \ A001694 = A106543 \ A052486 = A000027 \ A000961 \ A001694.

This sequence is A106543 \ A052486.

Example

a(1) = 6 since 6 is the smallest term in A024619 and is not powerful.
36 = 2^2 * 3^2 is not a term, though it is not a prime power, because all of its prime power factor exponents exceed 1.

Mathematica

nn = 120; Complement[Select[Range[nn], Not @* PrimePowerQ], Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}] ]

Prog

(Python)
from math import isqrt
from sympy import integer_nthroot, factorint, primepi
def A389065(n):
    def f(x): return int(n+sum(isqrt(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))+primepi(x))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 25 2025

Crossrefs

Cf. A000961, A001694, A002808, A024619, A052485, A052486, A106543, A120944, A126706, A286708, A332785.

Sequence in context: A323304 A325411 A106543 * A386207 A324455 A387721

Adjacent sequences: A389062 A389063 A389064 * A389066 A389067 A389068

Keyword

nonn,easy

Author

Michael De Vlieger, Sep 22 2025

Status

approved