A058061 - OEIS

%I #24 Jan 15 2024 01:48:05

%S 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,

%T 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,

%U 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

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

%C From _Bernard Schott_, Mar 24 2020: (Start)

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

%C 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)

%H Michael De Vlieger, <a href="/A058061/b058061.txt">Table of n, a(n) for n = 1..10000</a>

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

%F Additive with a(p^e) = A001222(e+1). - _Amiram Eldar_, Jan 15 2024

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

%t Table[PrimeOmega@ DivisorSigma[0, n], {n, 120}] (* _Michael De Vlieger_, Feb 18 2017 *)

%o (PARI) a(n) = bigomega(numdiv(n)); \\ _Michel Marcus_, Dec 14 2013

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

%K nonn,easy

%O 1,6

%A _Labos Elemer_, Nov 23 2000