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URL: https://oeis.org/A389065

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A389065
Numbers that are neither prime powers nor powerful.
5
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
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
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Sep 22 2025
STATUS
approved