A070012 -id:A070012 - cp's OEIS frontend

A088529 Numerator of Bigomega(n)/Omega(n).

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

Offset: 2

Cino Hilliard, Nov 16 2003

			bigomega(24) / omega(24) = 4/2 = 2, so a(24) = 2.

H. Z. Cao, On the average of exponents, Northeast. Math. J., Vol. 10 (1994), pp. 291-296.

Antti Karttunen, Table of n, a(n) for n = 2..10000
R. L. Duncan, On the factorization of integers, Proc. Amer. Math. Soc. 25 (1970), 191-192.
Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001222, A070012, A070013, A070014, A088530 (gives the denominator).

Cf. A136141, A272531.

Table[Numerator[PrimeOmega[n]/PrimeNu[n]], {n, 2, 100}] (* Michael De Vlieger, Jul 12 2017 *)

for(x=2,100,y=bigomega(x)/omega(x);print1(numerator(y)","))

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

Let B = number of prime divisors of n with multiplicity, O = number of distinct prime divisors of n. Then a(n) = numerator of B/O.
a(n) = A136565(n) = A181591(n) for n: 2 <= n < 24. - Reinhard Zumkeller, Nov 01 2010
Sum_{k=2..n} a(k)/A088530(k) ~ n + O(n/log(log(n))) (Duncan, 1970). - Amiram Eldar, Oct 14 2022
Sum_{k=2..n} a(k)/A088530(k) = n + c_1 * n/log(log(n)) + c_2 * n/log(log(n))^2 + O(n/log(log(n))^3), where c_1 = A136141 and c_2 = A272531 (Cao, 1994; Finch, 2020). - Amiram Eldar, Dec 15 2022

A088530 Denominator of bigomega(n)/omega(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1

Offset: 2

Cino Hilliard, Nov 16 2003

a(n) is the denominator of A022559(n)/A000720(n). - Robert Israel, Jan 08 2024

			bigomega(24) / omega(24) = 4/2 = 2/1, so a(24) = 1.

Antti Karttunen, Table of n, a(n) for n = 2..10000
R. L. Duncan, On the factorization of integers, Proc. Amer. Math. Soc. 25 (1970), 191-192.
Index entries for sequences computed from exponents in factorization of n

Cf. A001221, A001222, A000720, A022559, A070012, A070013, A070014, A088529 (gives the numerator).

N:= 100:
W:= ListTools:-PartialSums(map(numtheory:-bigomega,[$1..N])):
seq(denom(W[i]/numtheory:-pi(i)),i=2..N); # Robert Israel, Jan 08 2024

Table[Denominator[PrimeOmega[n]/PrimeNu[n]],{n,2,100}] (* Harvey P. Dale, Mar 22 2012 *)

for(x=2,100,y=bigomega(x)/omega(x);print1(denominator(y)","))

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

Let B = number of prime divisors of n with multiplicity, O = number of distinct prime divisors of n. Then a(n) = denominator of B/O.

A070011 Numbers n such that number of prime factors divided by the number of distinct prime factors is not an integer.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 108, 112, 116, 117, 120, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 168, 171, 172, 175, 176, 180, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 242

Offset: 1

Rick L. Shepherd, Apr 11 2002

nonn

			45 is a term because 45 = 3^2 * 5 gives bigomega(45)=3 and omega(45)=2 and 3/2 is not an integer.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A067340 (complement), A070012 (floor(bigomega(n)/omega(n))).

Different from A084679.

Select[Range[2,1000],!IntegerQ[PrimeOmega[#]/PrimeNu[#]]&] (* Enrique Pérez Herrero, Dec 20 2012 *)

v=[]; for(n=2,300,if(denominator(bigomega(n)/omega(n))<>1,v=concat(v,n))); v

is(n)=my(f=factor(n)[,2]); n>9 && vecsum(f)%#f!=0 \\ Charles R Greathouse IV, Oct 16 2015

A001222(n)/A001221(n) (i.e. bigomega(n)/omega(n)) is not an integer.

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

Original entry on oeis.org

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

Rick L. Shepherd, Apr 11 2002

nonn

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

			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.

G. C. Greubel, Table of n, a(n) for n = 2..1000

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)).

Table[Round[PrimeOmega[n]/PrimeNu[n]], {n, 2, 50}] (* G. C. Greubel, May 08 2017 *)

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

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

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

Original entry on oeis.org

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

Rick L. Shepherd, Apr 11 2002

nonn

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

			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.

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

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.

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

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

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

(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

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

A289621 Compound filter (omega & bigomega): a(1) = 0, for n > 1, a(n) = P(A001221(n), A001222(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 4, 2, 5, 1, 8, 1, 5, 5, 7, 1, 8, 1, 8, 5, 5, 1, 12, 2, 5, 4, 8, 1, 13, 1, 11, 5, 5, 5, 12, 1, 5, 5, 12, 1, 13, 1, 8, 8, 5, 1, 17, 2, 8, 5, 8, 1, 12, 5, 12, 5, 5, 1, 18, 1, 5, 8, 16, 5, 13, 1, 8, 5, 13, 1, 17, 1, 5, 8, 8, 5, 13, 1, 17, 7, 5, 1, 18, 5, 5, 5, 12, 1, 18, 5, 8, 5, 5, 5, 23, 1, 8, 8

Offset: 1

Antti Karttunen, Jul 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000027, A046523.

Cf. A001221, A001222, A008966, A046660, A070012, A070013, A070014, A088529, A088530, A181591 (sequences with matching equivalence classes).

A289621(n) = if(1==n,0,(1/2)*(2 + ((omega(n)+bigomega(n))^2) - omega(n) - 3*bigomega(n)));

(define (A289621 n) (if (= 1 n) 0 (* (/ 1 2) (+ (expt (+ (A001221 n) (A001222 n)) 2) (- (A001221 n)) (- (* 3 (A001222 n))) 2))))

a(1) = 0, for n > 1, a(n) = (1/2)*(2 + ((A001221(n)+A001222(n))^2) - A001221(n) - 3*A001222(n)).

A382266 Numerator of the harmonic mean of the exponents in the prime factorization of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 3, 2, 1, 3, 4, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 4, 1, 1, 8, 2, 4, 1, 4, 1, 3, 1, 3, 1, 1, 1, 6, 1, 1, 4, 6, 1, 1, 1, 4, 1, 1, 1, 12, 1, 1, 4, 4, 1, 1, 1, 8, 4, 1, 1, 6, 1, 1, 1, 3, 1, 6, 1, 4, 1, 1, 1, 5, 1, 4, 4, 2

Offset: 2

Ilya Gutkovskiy, Mar 19 2025

			1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4/3, 1, 1, 1, 4, 1, 4/3, 1, 4/3, 1, 1, 1, 3/2, 2, 1, 3, 4/3, ...

Index entries for sequences computed from exponents in factorization of n.

Cf. A070012, A088529, A250096, A382267 (denominators).

a:= n-> (l-> numer(nops(l)/add(1/i[2], i=l)))(ifactors(n)[2]):
seq(a(n), n=2..100);  # Alois P. Heinz, Mar 21 2025

Table[HarmonicMean[(#[[2]] & /@ FactorInteger[n])], {n, 2, 100}] // Numerator

A382267 Denominator of the harmonic mean of the exponents in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 5, 1, 3, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 2, 1, 5, 1, 3, 1, 1, 1, 3, 1, 3, 3, 1

Offset: 2

Ilya Gutkovskiy, Mar 19 2025

			1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4/3, 1, 1, 1, 4, 1, 4/3, 1, 4/3, 1, 1, 1, 3/2, 2, 1, 3, 4/3, ...

Index entries for sequences computed from exponents in factorization of n.

Cf. A070012, A088530, A250097, A382266 (numerators).

a:= n-> (l-> denom(nops(l)/add(1/i[2], i=l)))(ifactors(n)[2]):
seq(a(n), n=2..100);  # Alois P. Heinz, Mar 21 2025

Table[HarmonicMean[(#[[2]] & /@ FactorInteger[n])], {n, 2, 100}] // Denominator

cp's OEIS Frontend

A088529 Numerator of Bigomega(n)/Omega(n).

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A088530 Denominator of bigomega(n)/omega(n).

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A070011 Numbers n such that number of prime factors divided by the number of distinct prime factors is not an integer.

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

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

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A289621 Compound filter (omega & bigomega): a(1) = 0, for n > 1, a(n) = P(A001221(n), A001222(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A382266 Numerator of the harmonic mean of the exponents in the prime factorization of n.

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