A070014 - OEIS

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

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OFFSET

2,3

COMMENTS

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 2..10000

Index entries for sequences computed from exponents in factorization of n

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

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

(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

(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

CROSSREFS

Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is 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.

Sequence in context: A371148 A088388 A070013 * A051903 A324912 A157754

Adjacent sequences: A070011 A070012 A070013 * A070015 A070016 A070017

KEYWORD

nonn

AUTHOR

Rick L. Shepherd, Apr 11 2002

STATUS

approved