OFFSET
1,2
COMMENTS
The "excess" of a number is the number of prime divisors with multiplicity (the Omega function, A001222) minus the number of distinct prime divisors (the omega function, A001221). A046660(n) gives the excess of n.
Since squarefree numbers have no excess, this sequence is essentially A046660 with the 0's removed.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
FORMULA
Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p-1)) / (1-6/Pi^2) = A136141/A229099 = 1.9719717... - Amiram Eldar, Feb 10 2021
EXAMPLE
Since 16 = 2^4, 16 has four prime divisors, but only one distinct divisor. Hence Omega(16) - omega(16) = 4 - 1 = 3. As 16 is the fifth number that is not squarefree, its corresponding 3 is a(5) in this sequence.
17 is prime and thus has no excess and no corresponding term in this sequence.
18 = 2 * 3^2, Omega(18) - omega(18) = 3 - 2 = 1, thus a(6) = 1.
MATHEMATICA
DeleteCases[Table[PrimeOmega[n] - PrimeNu[n], {n, 200}], 0] (* Alonso del Arte, Aug 05 2016 *)
PROG
(PARI) for(n=1, 200, if(bigomega(n)!=omega(n), print1(bigomega(n)-omega(n), ", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Fröhlich, Aug 05 2016
STATUS
approved