Omega-rad resonance numbers

Numbers m such that rad(m) * omega(m) = m.

Created: 6/27/2026 | Author: Vincenzo Manto and James C. McMahon , Jun 19 2026

This sequence has been cited by and influenced 6 other sequences

Formal Definition

Numbers n such that:

n = rad(n) * ω(n)

Structural Analysis & Motivation

This is the nearest sibling of A396594. The designation "Omega-rad resonance numbers" stems from a structural resonance between the radical of a number and its distinct prime factor count. The radical rad(n) captures the unique prime factors, while ω(n) counts the distinct prime factors. 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) * omega(p) = p * 1 = p.
Prime powers p^k (k > 1) are never terms since p * 1 != p^k.
Composite terms classified by r = omega(m):
- r = 2: m = 4*p for odd primes p (e.g., 12, 20, 28, 44).
- r = 3: m = 9*p*q for distinct primes p, q != 3 (e.g., 90).
- r = 4: m = 8*p*q*s for distinct odd primes p, q, s (e.g., 840).

Sequence Chart

Data

2,3,5,7,11,12,13,17,19,20,23,28,29,31,37,41,43,44,47,52,53,59,61,67,68,71,73,76,79,83,89,90,92,97,101,103,107,109,113,116,124,126,127,131,137,139,148,149,151,157,163,164,167,172,173,179,181,188,191,193,197,198,199

Computational Implementations

PYTHON

from math import prod
from sympy import primefactors
def ok(m): return len(f := primefactors(m)) * prod(f) == m
print([m for m in range(1, 200) if ok(m)])

Mathematica

q[m_] := Module[{p = FactorInteger[m][[;; , 1]]}, Length[p] * Times @@ p == m]; q[1] = False; Select[Range[200], q] (* _Amiram Eldar_, Jun 21 2026 *)

Cross-References

See also OEIS entries: Cf A000040 A001221 A007947 A065512