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URL: https://oeis.org/A385113

⇱ A385113 - OEIS

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A385113
a(1) = 0, a(n) = 1 + 2*[omega(n) > 1] + [bigomega(n) > omega(n)], Iverson brackets, where omega = A001221 and bigomega = A001222.
3
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
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
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Sep 17 2025
STATUS
approved