1,1

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).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Paul Erdős and Mark Kac, The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions, American Journal of Mathematics, Vol. 62, No. 1 (1940), pp. 738-742, alternative link.

Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, p. 75.

Alfréd Rényi and Pál Turán, On a theorem of Erdős-Kac, Acta Arithmetica, Vol. 4, No. 1 (1958), pp. 71-84.

Eric Weisstein's World of Mathematics, Erdős-Kac theorem.

Eric Weisstein's World of Mathematics, Erfc.

Wikipedia, Erdős-Kac theorem.

Select[Range[3, 10^5], PrimeNu[#] > Log[Log[#]] + 2 * Sqrt[Log[Log[#]]] &]

Cf. A001221, A239383, A336909.

Sequence in context: A286978 A293086 A293099 * A286715 A365706 A081176

Adjacent sequences: A336907 A336908 A336909 * A336911 A336912 A336913

Amiram Eldar, Aug 07 2020

approved