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

Author

Labos Elemer, Nov 23 2000

Keywords

• nonn
• easy

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)

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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

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