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a(n) = Sum_{k = 1..10^n} number of primes (counted with multiplicity) dividing k (A001222).
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%I #31 Oct 12 2024 03:54:41

%S 0,15,239,2877,31985,343614,3626619,37861249,392351272,4044220058,

%T 41518796555,424904645958,4337589196099,44189168275565,

%U 449411845856902,4564053529871328,46294122513328879,469075734968975581,4748553675150670580,47797839092868715542

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

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

%F From _Amiram Eldar_, Oct 11 2024: (Start)

%F a(n) = A022559(10^n).

%F a(n) ~ 10^n * (log(log(10^n)) + B_2), where B_2 = A083342. (End)

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

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

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

%o (PARI) g(n) = for(x=0,n,print1(bigomega((10^x)!),",")) \\ _Cino Hilliard_, Jul 04 2007

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

%K nonn

%O 0,2

%A _Rick L. Shepherd_, Jun 07 2002

%E a(9) from _Charles R Greathouse IV_, Dec 11 2008

%E a(11)-a(12) from _Giovanni Resta_, Oct 26 2012

%E a(13)-a(17) from _Hiroaki Yamanouchi_, Aug 28 2014

%E a(18)-a(19) from _Henri Lifchitz_, Dec 17 2017