A070014 Ceiling 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, 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, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1

Offset: 2

Author

Rick L. Shepherd, Apr 11 2002

Keywords

• nonn

Comments

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

Examples

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

Links

• Antti Karttunen, Table of n, a(n) for n = 2..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is an integer before ceil is applied), A070011 (ratio is not an integer), A070012 (floor of ratio), A070013 (ratio rounded), A046660 (bigomega(n)-omega(n)), A088529, A088530.

Programs

Mathematica

Table[Ceiling[PrimeOmega[n]/PrimeNu[n]], {n, 2, 106}] (* Michael De Vlieger, Jul 12 2017 *)

PARI

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

Python

from sympy import primefactors, ceiling
def bigomega(n): return 0 if n==1 else bigomega(n//primefactors(n)[0]) + 1
def omega(n): return len(primefactors(n))
def a(n): return ceiling(bigomega(n)/omega(n))
print([a(n) for n in range(2, 51)]) # Indranil Ghosh, Jul 13 2017

Scheme

(define (A070014 n) (let ((a (A001222 n)) (b (A001221 n))) (if (zero? (modulo a b)) (/ a b) (+ 1 (/ (- a (modulo a b)) b))))) ;; Antti Karttunen, Jul 12 2017

Formula

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