A070012 Floor of number of prime factors of n divided by the number of n's distinct prime factors.

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

Offset: 2

Author

Rick L. Shepherd, Apr 11 2002

Keywords

• nonn

Comments

a(n) is the integer part of the average of the exponents in the prime factorization of n.

Examples

			a(12)=1 because 12=2^2 * 3^1 and floor(bigomega(12)/omega(12)) = floor((2+1)/2) = 1.
a(36)=2 because 36=2^2 * 3^2 and floor(bigomega(36)/omega(36)) = floor((2+2)/2) = 2.
a(60)=1 because 60=2^2 * 3^1 * 5^1 and floor(bigomega(60)/omega(60)) = floor((2+1+1)/3) = 1.
36 is in A067340. 12 and 60 are in A070011.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..5000

Crossrefs

Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is an integer before floor applied), A070011 (ratio is not an integer), A070013 (ratio rounded), A070014 (ceiling of ratio), A046660 (bigomega(n)-omega(n)).

Programs

Mathematica

A070012[n_]:=Floor[PrimeOmega[n]/PrimeNu[n]];Array[A070012,100]

PARI

v=[]; for(n=2,150,v=concat(v,(bigomega(n)\omega(n)))); v

Formula

a(n) = floor(bigomega(n)/omega(n)) for n>=2.