Vincenzo Manto

A397574

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

Created: 7/8/2026 | Author: Vincenzo Manto and James C. McMahon , Jul 01 2026

This sequence has been cited by and influenced 1 other sequence

Official Comments

A term m > 1 is in the sequence if and only if the exponent of every prime factor in its prime factorization is exactly equal to omega(m). Thus, m must be of the form (p1 * p2 * ... * pk)^k for distinct primes pi.
Terms are mostly primes: for any p, omega(p) = 1 and rad(p) = p, so rad(p)^omega(p) = p ^ 1.
Except for 1 and the prime numbers (A000040), all terms are perfect powers.
Composite terms are the k-th powers of squarefree numbers having exactly k prime factors. For k = 2, these are the squares of squarefree semiprimes.

Sequence Chart

Graph of A397574

Data

1,2,3,5,7,11,13,17,19,23,29,31,36,37,41,43,47,53,59,61,67,71,73,79,83,89,97,100,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,196,197,199,211,223,225,227,229,233,239

Computational Implementations

PYTHON

from math import prod
from sympy import primefactors
def ok(m): return prod(f := primefactors(m)) ** len(f) == m
print([m for m in range(1, 250) if ok(m)])

CODE

isok(m) = {my(p = factor(m)[, 1]); m == vecprod(p)^#p;} \\ _Amiram Eldar_, Jul 02 2026

Mathematica

q[m_]:=(Times@@First/@FactorInteger[m])^PrimeNu[m]==m;Select[Range[239],q]

Cross-References

See also OEIS entries: Cf A000040 A001221 A007947 A397221