A397574 - OEIS

%I #11 Jul 07 2026 22:39:11

%S 1,2,3,5,7,11,13,17,19,23,29,31,36,37,41,43,47,53,59,61,67,71,73,79,

%T 83,89,97,100,101,103,107,109,113,127,131,137,139,149,151,157,163,167,

%U 173,179,181,191,193,196,197,199,211,223,225,227,229,233,239

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

%C 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.

%C Terms are mostly primes: for any p, omega(p) = 1 and rad(p) = p, so rad(p)^omega(p) = p ^ 1.

%C Except for 1 and the prime numbers (A000040), all terms are perfect powers.

%C 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.

%H James C. McMahon, <a href="/A397574/b397574.txt">Table of n, a(n) for n = 1..10000</a>

%e m = 36 is a term since rad(36)^omega(36) = 6 ^ 2 = 36.

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

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

%o (Python)

%o from math import prod

%o from sympy import primefactors

%o def ok(m): return prod(f := primefactors(m)) ** len(f) == m

%o print([m for m in range(1, 250) if ok(m)])

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

%Y Cf. A000040, A001221, A007947, A397221.

%K nonn

%O 1,2

%A _Vincenzo Manto_ and _James C. McMahon_, Jul 01 2026