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%I #8 Aug 09 2020 03:15:03
%S 3,2310,2730,30030,39270,43890,46410,51870,53130,60060,62790,66990,
%T 67830,71610,72930,78540,79170,81510,82110,84630,85470,87780,90090,
%U 91770,92820,94710,98670,99330,101010,102102,103530,103740,106260,106590,108570,110670,111930
%N Numbers k > 2 such that omega(k) > log(log(k)) + 2 * sqrt(log(log(k))), where omega(k) is the number of distinct primes dividing k (A001221).
%C According to Erdős-Kac theorem, the asymptotic density of the sequence of numbers k such that omega(k) > log(log(k)) + c * sqrt(log(log(k))), for all real numbers c, is erfc(c/sqrt(2))/2. Here c = 2, and therefore the asymptotic density of this sequence is erfc(sqrt(2))/2 = 0.022750... (A239383).
%H Amiram Eldar, <a href="/A336910/b336910.txt">Table of n, a(n) for n = 1..10000</a>
%H Paul Erdős and Mark Kac, <a href="https://doi.org/10.2307%2F2371483">The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions</a>, American Journal of Mathematics, Vol. 62, No. 1 (1940), pp. 738-742, <a href="https://users.renyi.hu/~p_erdos/1940-12.pdf">alternative link</a>.
%H Mark Kac, <a href="http://www.gibbs.if.usp.br/~marchett/estocastica/MarkKac-Statistical-Independence.pdf">Statistical Independence in Probability, Analysis and Number Theory</a>, Carus Monograph 12, Math. Assoc. Amer., 1959, p. 75.
%H Alfréd Rényi and Pál Turán, <a href="https://eudml.org/doc/206110">On a theorem of Erdős-Kac</a>, Acta Arithmetica, Vol. 4, No. 1 (1958), pp. 71-84.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erdos-KacTheorem.html">Erdős-Kac theorem</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Erfc.html">Erfc</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem">Erdős-Kac theorem</a>.
%t Select[Range[3, 10^5], PrimeNu[#] > Log[Log[#]] + 2 * Sqrt[Log[Log[#]]] &]
%Y Cf. A001221, A239383, A336909.
%K nonn
%O 1,1
%A _Amiram Eldar_, Aug 07 2020
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