A058061 Number of prime factors (counted with multiplicity) of d(n), the number of divisors of n.

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 3, 3

Offset

1,6

Comments

From Bernard Schott, Mar 24 2020: (Start)

a(n) = 1 iff n = p^(q-1) with p, q primes (A009087).

a(n) = 2 if n=p*q with p, q primes (A006881), or if n=p^2*q with p, q primes (A054753), or if n=p^4*q with p, q primes (A178739), or if n=p^6*q with p, q primes (A189987), or if n=p^2*q^4 with p, q primes (A189988), or if n=p^(m-1) with p prime and m is semiprime in A001358 (not exhaustive). (End)

Links

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

Formula

a(n) = A001222(A000005(n)).

Additive with a(p^e) = A001222(e+1). - Amiram Eldar, Jan 15 2024

Example

For n=120, d(120)=16, a(120)=4.

Mathematica

Table[PrimeOmega@ DivisorSigma[0, n], {n, 120}] (* Michael De Vlieger, Feb 18 2017 *)

Prog

(PARI) a(n) = bigomega(numdiv(n)); \\ Michel Marcus, Dec 14 2013

Crossrefs

Cf. A001222, A000005, A058060, A079057 (partial sums).

Sequence in context: A160980 A065031 A305832 * A371090 A064547 A318306

Adjacent sequences: A058058 A058059 A058060 * A058062 A058063 A058064

Keyword

nonn,easy

Author

Labos Elemer, Nov 23 2000

Status

approved