A305054 If n = Product_i prime(x_i)^k_i, then a(n) = Sum_i k_i * omega(x_i), where omega = A001221 is number of distinct prime factors.

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 1, 0, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 1, 2, 2, 1, 2, 2, 1, 3, 0, 3, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 1, 4, 1, 1, 2, 2, 2, 3

Offset: 1

Author

Gus Wiseman, May 24 2018

Keywords

• nonn

Crossrefs

Cf. A001221, A003963, A056239, A076078, A112798, A286520, A290103, A302242, A304714, A304716, A305052, A305053.

Programs

Mathematica

Table[If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*PrimeNu[PrimePi[p]]]],{n,100}]

PARI

a(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,2]*omega(primepi(f[k,1])));} \\ Michel Marcus, Jun 09 2018

Formula

Totally additive with a(prime(n)) = omega(n).

a(n) = A305053(n) + A001221(n). - Michel Marcus, Jun 09 2018