A070013 Number of prime factors of n divided by the number of n's distinct prime factors (rounded).

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

Offset

2,3

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 2..1000 (corrected by Sean A. Irvine)

Formula

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

Example

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

Mathematica

Table[If[Ceiling[#] - # == 1/2, Ceiling[#], Round[#] ] &[PrimeOmega[n]/PrimeNu[n] ], {n, 2, 50}] (* G. C. Greubel, May 08 2017, corrected by Michael De Vlieger, Mar 02 2026 *)

Prog

(PARI) a(n) = round(bigomega(n)/omega(n));

Crossrefs

Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is an integer before rounding), A070011 (ratio is not an integer), A070012 (floor of ratio), A070014 (ceiling of ratio), A046660 (bigomega(n)-omega(n)).

Sequence in context: A362333 A371148 A088388 * A070014 A051903 A324912

Adjacent sequences: A070010 A070011 A070012 * A070014 A070015 A070016

Keyword

nonn

Author

Rick L. Shepherd, Apr 11 2002

Status

approved