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A070014 Ceiling of number of prime factors of n divided by the number of n's distinct prime factors. 6

%I #22 Dec 30 2023 12:35:51

%S 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,

%T 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,

%U 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

%N Ceiling of number of prime factors of n divided by the number of n's distinct prime factors.

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

%H Antti Karttunen, <a href="/A070014/b070014.txt">Table of n, a(n) for n = 2..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

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

%e 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.

%t Table[Ceiling[PrimeOmega[n]/PrimeNu[n]], {n, 2, 106}] (* _Michael De Vlieger_, Jul 12 2017 *)

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

%o (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

%o (Python)

%o from sympy import primefactors, ceiling

%o def bigomega(n): return 0 if n==1 else bigomega(n//primefactors(n)[0]) + 1

%o def omega(n): return len(primefactors(n))

%o def a(n): return ceiling(bigomega(n)/omega(n))

%o print([a(n) for n in range(2, 51)]) # _Indranil Ghosh_, Jul 13 2017

%Y 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.

%K nonn

%O 2,3

%A _Rick L. Shepherd_, Apr 11 2002

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