A385113 a(1) = 0, a(n) = 1 + 2*[omega(n) > 1] + [bigomega(n) > omega(n)], Iverson brackets, where omega = A001221 and bigomega = A001222.

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, 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, 3, 3, 1, 4, 1, 3, 4, 4, 3, 3, 1, 4, 2, 3, 1, 4, 3, 3, 3, 4, 1

Offset

1,4

Comments

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

For prime n, a(n) = 1.

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

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

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

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

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

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

Example

                   Prime
Class     Example  power? Squarefree? a(n) Category
-----------------------------------------------------------------------------
n in A000040    7  True   True          1  primes
n in A246547    8  True   False         2  proper prime powers
n in A120944   10  False  True          3  squarefree composites
n in A126706   12  False  False         4  neither squarefree nor prime power

Mathematica

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

Prog

(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
(SageMath)
def A385113(n: int) -> int:
    return 4 - 2*is_prime_power(n) - is_squarefree(n) if n > 1 else 0
print([A385113(n) for n in range(1, 90)])  # Peter Luschny, Sep 18 2025

Crossrefs

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

Sequence in context: A303757 A182850 A323014 * A391024 A350331 A293227

Adjacent sequences: A385110 A385111 A385112 * A385114 A385115 A385116

Keyword

nonn,easy

Author

Michael De Vlieger, Sep 17 2025

Status

approved