A397574 Numbers m such that rad(m)^omega(m) = m.

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

Offset

1,2

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 (p_1 * p_2 * ... * p_k)^k for distinct primes p_i.

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.

Links

James C. McMahon, Table of n, a(n) for n = 1..10000

Example

m = 36 is a term since rad(36)^omega(36) = 6 ^ 2 = 36.
m = 8 is not a term since rad(8)^omega(8) = 2 ^ 1 = 2 != 8.

Mathematica

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

Prog

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

Crossrefs

Cf. A000040, A001221, A007947, A397221.

Sequence in context: A350342 A385402 A369093 * A171019 A249825 A117095

Adjacent sequences: A397565 A397567 A397568 * A397575 A397576 A397577

Keyword

nonn,new

Author

Vincenzo Manto and James C. McMahon, Jul 01 2026

Status

approved