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

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

Offset: 0

Author

Rick L. Shepherd, Jun 07 2002

Keywords

• nonn

Comments

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

Examples

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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)

Extensions

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