A391024 Classification of numbers predicated according to prime powers, squarefree, powerful, and perfect powers.

0, 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, 6, 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, 5, 1, 3, 4, 4, 3, 3, 1, 4, 2, 3, 1, 4, 3, 3

Offset

0,5

Comments

Let Q be the set of prime factor exponents of k.

Let omega = A001221 and let bigomega = A001222.

Class Description

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0 {0, 1}.

1 A000040 {p : 1 = omega(p) = bigomega(p)}, primes.

2 A246547 {k : 1 = omega(k) < bigomega(k)}, proper prime powers.

3 A120944 {k : 1 < omega(k) = bigomega(k)}, squarefree composites.

4 A332785 {k : 1 < omega(k) < bigomega(k), min(Q) = 1},

nonsquarefree weak numbers.

5 A052486 {k : 1 < omega(k) < bigomega(k), min(Q) > 1, gcd(Q) = 1}

Achilles numbers, powerful but not perfect powers.

6 A131605 {k : 1 < omega(k) < bigomega(k), gcd(Q) > 1},

perfect powers that are not prime powers.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Michael De Vlieger, Table of n = 0..1295 arranged in rows of 36 terms, with a color code that represents a(n) as follows: a(n) = 0 in gray, a(n) = 1 in red, a(n) = 2 in gold, a(n) = 3 in green, a(n) = 4 in blue, a(n) = 5 in purple, and a(n) = 6 in magenta.

Example

a(n) > 1 for composite n (in A002808).
a(n) > 2 for n that is not a prime power (in A024619).
a(n) > 3 for n that is neither squarefree nor prime power (in A126706).
a(n) > 4 for n that is powerful and not a prime power (in A286708).
a(n) > 5 for n that is a perfect power that is not a prime power (in A131605).
a(n) < 3 for prime power n (in A000961).
a(n) is in {0, 1, 3} for squarefree n (in A005117).
a(n) is in {2, 4, 5, 6} for nonsquarefree n (in A013929).
a(n) is in {0, 2, 6} for perfect power n (in A001597).
a(n) is in {0, 2, 5, 6} for powerful n (in A001694).
a(n) is in {1, 3, 4, 5} for n in A007916.
a(n) is in {1, 3, 4} for weak n (in A052485).
Therefore, a(5) = 1, a(8) = 2, a(10) = 3, a(18) = 4, a(72) = 5, and a(36) = 6.

Mathematica

f[n_] :=
  If[n < 2, 0,
    If[# == 4,
      Which[And[Min[#] > 1, GCD @@ # > 1], 6,
        Min[#] > 1, 5,
        True, 4] &@ FactorInteger[n][[;; , -1]], #] &[
  1 + 2*Boole[PrimeNu[n] > 1] + Boole[PrimeOmega[n] > PrimeNu[n] ] ] ];
Array[f, 120, 0]

Crossrefs

Cf. A000040, A000961, A001597, A001694, A002808, A005117, A007916, A013929, A024619, A120944, A126706, A131605, A246547, A286708, A332785, A385113.

Sequence in context: A182850 A323014 A385113 * A350331 A293227 A291208

Adjacent sequences: A391021 A391022 A391023 * A391025 A391026 A391027

Keyword

nonn,easy,changed

Author

Michael De Vlieger, Dec 03 2025

Status

approved