Official Comments

Numbers m such that bigomega(m) - omega(m) = core(m).
In other words, numbers m such that A001222(m) - A001221(m) = A046660(m) = A007913(m).
Squarefree numbers are naturally excluded since for any squarefree number m >= 1, bigomega(m) - omega(m) = 0, while core(m) = m >= 1.
The difference bigomega(m) - omega(m) is sometimes called the prime factor multiplicity index of m (A046660).
The set of prime squares A001248 is a proper subset since A046660(p^2) = 1 and core(p^2) = 1.
Of the 56250 terms m <= 2^40 in A286708 (powerful but not a prime power), 56220 are Achilles numbers and 30 are in A386762 (perfect powers of numbers neither squarefree nor powerful). Most of the terms in the intersection with A386762 are cubes except 380204032 = 2^10 * 13^5 and 21924480357 = 3^10 * 13^5.

Sequence Chart

Data

Computational Implementations

from math import prod
from sympy import factorint
def ok(m): return m > 1 and sum((d := factorint(m)).values()) - len(d) == prod(p for p, e in d.items() if e % 2)
print([m for m in range(1, 8000) if ok(m)])

isok(k) = my(f=factor(k)); core(f) == bigomega(f) - omega(f); \\ _Michel Marcus_, Jul 10 2026

core[x_] := Sqrt[x] /. (c_ : 1)*a_^(b_ : 0) :> (c*a^b)^2; Select[Range[8000], PrimeOmega[#] - PrimeNu[#] == core[#] &]

Cross-References

See also OEIS entries: Cf A001222 A001221 A007913 A396594 A397221 A046660 A001248