A275699 Excess of numbers that are not squarefree.

1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 1, 3, 1, 1, 3, 3, 1, 2, 1, 1, 4, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 1, 6, 1, 2, 2, 1, 4, 1, 1, 1, 2, 1, 1, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 5, 2, 1, 3, 1, 1, 3, 1, 4, 1, 4, 2, 1

Offset: 1

Author

Felix Fröhlich, Aug 05 2016

Keywords

• nonn

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.

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A001221, A001222, A013929, A046660, A136141, A229099.

Programs

Mathematica

DeleteCases[Table[PrimeOmega[n] - PrimeNu[n], {n, 200}], 0] (* Alonso del Arte, Aug 05 2016 *)

PARI

for(n=1, 200, if(bigomega(n)!=omega(n), print1(bigomega(n)-omega(n), ", ")))

Formula

a(n) = A046660(A013929(n)).

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