A083342 -id:A083342 - cp's OEIS frontend

A360729 a(n) is the number of prime factors of the n-th powerful number (counted with repetition).

0, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 5, 4, 4, 5, 2, 3, 7, 6, 2, 4, 5, 6, 4, 5, 8, 7, 2, 6, 3, 2, 5, 6, 7, 4, 4, 5, 9, 2, 8, 4, 7, 5, 4, 6, 6, 7, 2, 8, 6, 2, 5, 7, 6, 10, 4, 5, 9, 4, 4, 8, 5, 3, 5, 2, 5, 4, 4, 7, 8, 2, 9, 6, 7, 2, 6, 8, 7, 6, 11, 4, 7, 3, 2, 10, 5

Offset: 1

Amiram Eldar, Feb 18 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

Cf. A001222, A001694, A083342, A090699.

Similar sequences: A072047, A076399.

PrimeOmega[Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]

apply(bigomega, select(ispowerful, [1..3000]))

a(n) = A001222(A001694(n)).

Sum_{A001694(k) < x} a(k) = (2*zeta(3/2)/zeta(3))*sqrt(x)*log(log(x)) + (2*(B_2 - log(2)) + Sum_{p prime} (3/((p^(3/2)+1))))*(zeta(3/2)/zeta(3))*sqrt(x) + O(sqrt(x)/sqrt(log(x))), where B_2 = A083342 (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

A091589 Decimal expansion of the analog of the Mertens constant B_2 in the asymptotic series for the variance of the number of prime factors Omega.

Original entry on oeis.org

7, 6, 4, 7, 8, 4, 8, 0, 9, 7, 9, 7, 8, 4, 6, 7, 6, 3, 0, 6, 4, 6, 3, 2, 2, 6, 2, 3, 4, 1, 0, 5, 2, 7, 6, 4, 6, 3, 0, 9, 5, 8, 0, 4, 3, 3, 1, 2, 9, 0, 4, 2, 7, 5, 4, 7, 6, 7, 0, 1, 3, 8, 7, 8, 0, 2, 8, 6, 1, 7, 8, 2, 1, 2, 4, 9, 5, 5, 9, 8, 9, 1, 4, 4, 2, 2, 0, 1, 4, 2, 0, 5, 4, 3, 9, 6, 8, 7, 8, 6

Offset: 0

Eric W. Weisstein, Jan 22 2004

			0.76478480...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 95.

Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001222, A013661 (Pi^2/6), A086242, A083342.

Equals A083342 minus Pi^2/6 plus A086242. - R. J. Mathar, Oct 14 2010

Equals lim_{m->oo} ((1/m) * Sum_{k=1..m} A001222(k)^2 - ((1/m) * Sum_{k=1..m} A001222(k))^2 - log(log(m))). - Amiram Eldar, Jan 09 2024

More digits from R. J. Mathar, Oct 14 2010

More digits (using terms in A083342 and A086242) from Vaclav Kotesovec, Aug 12 2019

A363013 a(n) is the number of prime factors (counted with multiplicity) of the n-th cubefull number (A036966).

Original entry on oeis.org

0, 3, 4, 3, 5, 6, 4, 3, 7, 6, 5, 8, 3, 7, 9, 4, 7, 6, 8, 6, 10, 8, 3, 9, 8, 7, 11, 7, 3, 4, 9, 6, 5, 6, 10, 9, 8, 12, 3, 7, 10, 7, 9, 8, 3, 11, 10, 9, 13, 6, 8, 7, 11, 6, 8, 10, 3, 12, 4, 11, 6, 10, 14, 5, 7, 10, 6, 7, 9, 9, 12, 7, 9, 11, 3, 8, 9, 13, 7, 4, 3

Offset: 1

Amiram Eldar, May 13 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

Cf. A001222, A036966, A362974.

Similar sequences: A072047, A076399, A360729.

PrimeOmega[Select[Range[10000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

iscubefull(n) = n==1 || vecmin(factor(n)[, 2]) > 2;
apply(bigomega, select(iscubefull, [1..10000]))

a(n) = A001222(A036966(n)).
a(n) >= 3, for n > 1.
Sum_{A036966(k) < x} a(k) = 3*c*x^(1/3)*log(log(x)) + (3*(B_2 - log(2)) + Sum_{p prime} ((4*p^(1/3)+5)/(p^(5/3)+p^(1/3)+1)))*c*x^(1/3) + O(x^(1/3)/sqrt(log(x))), where B_2 = A083342 and c = A362974 (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

A071811 a(n) = Sum_{k = 1..10^n} number of primes (counted with multiplicity) dividing k (A001222).

Original entry on oeis.org

0, 15, 239, 2877, 31985, 343614, 3626619, 37861249, 392351272, 4044220058, 41518796555, 424904645958, 4337589196099, 44189168275565, 449411845856902, 4564053529871328, 46294122513328879, 469075734968975581, 4748553675150670580, 47797839092868715542

Offset: 0

Rick L. Shepherd, Jun 07 2002

nonn

Also bigomega( (10^n)! ), where bigomega(x): number of prime divisors of x, counted with multiplicity. - Cino Hilliard, Jul 04 2007

			a(1) = 15 because bigomega(1) + bigomega(2) + ... + bigomega(10) = 0+1+1+2+1+2+1+3+2+2 = 15.

Cf. A001222 (bigomega), A022559, A064182 (corresponding sums for distinct primes), A083342.

With[{s = Array[PrimeOmega, 10^6]}, {0}~Join~Array[Total@ Take[s, 10^#] &, Floor@ Log10@ Length@ s]] (* Michael De Vlieger, Dec 17 2017 *)

s=0; n=0; for(k=1,10^8, s=s+bigomega(k); if(k==10^n,print1(s,","); n++))

g(n) = for(x=0,n,print1(bigomega((10^x)!),",")) \\ Cino Hilliard, Jul 04 2007

From Amiram Eldar, Oct 11 2024: (Start)
a(n) = A022559(10^n).
a(n) ~ 10^n * (log(log(10^n)) + B_2), where B_2 = A083342. (End)

a(9) from Charles R Greathouse IV, Dec 11 2008
a(11)-a(12) from Giovanni Resta, Oct 26 2012
a(13)-a(17) from Hiroaki Yamanouchi, Aug 28 2014
a(18)-a(19) from Henri Lifchitz, Dec 17 2017

cp's OEIS Frontend

A360729 a(n) is the number of prime factors of the n-th powerful number (counted with repetition).

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A091589 Decimal expansion of the analog of the Mertens constant B_2 in the asymptotic series for the variance of the number of prime factors Omega.

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A363013 a(n) is the number of prime factors (counted with multiplicity) of the n-th cubefull number (A036966).

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A071811 a(n) = Sum_{k = 1..10^n} number of primes (counted with multiplicity) dividing k (A001222).

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