Bigomega-rad resonance numbers
Numbers m such that rad(m) * bigomega(m) = m.
Formal Definition
Structural Analysis & Motivation
This is the nearest sibling of A397221. This sequence comes from the same structural framework, replacing omega(k) with Ω(n). The designation "Bigomega-rad resonance numbers" stems from a structural resonance between the radical of a number and its total prime factor count. The radical rad(n) captures the unique prime factors, while Ω(n) counts all prime factors with multiplicity. The equality n = rad(n) * Ω(n) indicates a precise balance between these two measures.
Linked Articles
Together with M. De Vlieger, we wrote an extensive article on the structural properties of these numbers, generating a full family of related sequences (10+) and providing a comprehensive analysis of their distribution and properties. The article is available at this link.
Official Comments
Primes p are terms since rad(p) * bigomega(p) = p * 1 = p.
From Michael De Vlieger, Jul 03 2026: (Start)
Primes are the only squarefree numbers (in A005117) in the sequence. Squarefree k = rad(k) such that bigomega(k) = omega(k) > 2 do not appear in the sequence since k * omega(k) > k.
The only powerful number k (in A001694) in this sequence is 4 = 2^2 = 2*2, since powerful numbers k imply rad(k)^2 | k, and further, bigomega(k) >= k/rad(k), where k/rad(k) >= rad(k). Attempting to increment the left hand side only increases the ratio RHS/LHS, since RHS increases by a prime factor.
Consequences:
1. A175787 is a proper subset of this sequence.
2. There is no intersection of this sequence and A120944 (squarefree and composite).
3. {a(n)} \ A175787 is a proper subset of A332785.
4. 4 is the only perfect power (in A001597) and the only prime power (in A246547) in this sequence. (End)
Sequence Chart
Data
Cross-References
See also OEIS entries: Cf A000040 A001222 A001597 A001694 A007947 A175787 A332785 A397221