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A070014
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Ceiling of number of prime factors of n divided by the number of n's distinct prime factors.
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6
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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
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2,3
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COMMENTS
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a(n) is the ceiling of the average of the exponents in the prime factorization of n.
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 2..10000
Index entries for sequences computed from exponents in factorization of n
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FORMULA
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a(n) = ceiling(bigomega(n)/omega(n)) for n>=2.
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EXAMPLE
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a(12) = 2 because 12 = 2^2 * 3^1 and ceil(bigomega(12)/omega(12)) = ceil((2+1)/2) = 2. a(36) = 2 because 36 = 2^2 * 3^2 and ceil(bigomega(36)/omega(36)) = ceil((2+2)/2) = 2. a(60) = 2 because 60 = 2^2 * 3^1 * 5^1 and ceil(bigomega(60)/omega(60)) = ceil((2+1+1)/3) = 2. 36 is in A067340. 12 and 60 are in A070011.
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MATHEMATICA
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Table[Ceiling[PrimeOmega[n]/PrimeNu[n]], {n, 2, 106}] (* Michael De Vlieger, Jul 12 2017 *)
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PROG
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(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 map(a, range(2, 151)) # Indranil Ghosh, Jul 13 2017
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CROSSREFS
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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: A307907 A088388 A070013 * A051903 A324912 A157754
Adjacent sequences: A070011 A070012 A070013 * A070015 A070016 A070017
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KEYWORD
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nonn
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AUTHOR
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Rick L. Shepherd, Apr 11 2002
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STATUS
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approved
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