A385113 - OEIS

%I #25 Sep 27 2025 11:20:47

%S 0,1,1,2,1,3,1,2,2,3,1,4,1,3,3,2,1,4,1,4,3,3,1,4,2,3,2,4,1,3,1,2,3,3,

%T 3,4,1,3,3,4,1,3,1,4,4,3,1,4,2,4,3,4,1,4,3,4,3,3,1,4,1,3,4,2,3,3,1,4,

%U 3,3,1,4,1,3,4,4,3,3,1,4,2,3,1,4,3,3,3,4,1

%N a(1) = 0, a(n) = 1 + 2*[omega(n) > 1] + [bigomega(n) > omega(n)], Iverson brackets, where omega = A001221 and bigomega = A001222.

%C For empty product 1, a(1) = 0 by convention.

%C For prime n, a(n) = 1.

%C For proper prime power n = p^m, m > 1 (i.e., n in A246547) a(n) = 2.

%C For squarefree composite n (i.e., n in A120944) a(n) = 3.

%C For n that are neither squarefree nor composite (i.e., n in A126706), a(n) = 4.

%H Michael De Vlieger, <a href="/A385113/b385113.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 0, a(n) = 4 - 2*[is n a prime power?] - [is n squarefree?].

%F a(1) = 0, a(n) = 4 - 2*A010055(n) - A008966(n).

%F a(1) = 0, a(n) = A010051(n) + 2*A268340(n) + 3*A354819(n) + 4*A355447(n).

%e                    Prime

%e Class     Example  power? Squarefree? a(n) Category

%e -----------------------------------------------------------------------------

%e n in A000040    7  True   True          1  primes

%e n in A246547    8  True   False         2  proper prime powers

%e n in A120944   10  False  True          3  squarefree composites

%e n in A126706   12  False  False         4  neither squarefree nor prime power

%t {0}~Join~Table[1 + 2*Boole[PrimeNu[n] > 1] + Boole[PrimeOmega[n] > PrimeNu[n]], {n, 2, 120}]

%o (PARI) a(n) = if (n==1, 0, my(f=factor(n)); 1 + 2*(omega(f) > 1) + (bigomega(f) > omega(f))); \\ _Michel Marcus_, Sep 18 2025

%o (SageMath)

%o def A385113(n: int) -> int:

%o     return 4 - 2*is_prime_power(n) - is_squarefree(n) if n > 1 else 0

%o print([A385113(n) for n in range(1, 90)])  # _Peter Luschny_, Sep 18 2025

%Y Cf. A000040, A000941, A001221, A001222, A005117, A008966, A010551, A010055, A013929, A024619, A120944, A126706, A246655, A246547, A268340, A354819, A355447, A388303.

%K nonn,easy

%O 1,4

%A _Michael De Vlieger_, Sep 17 2025