A336910 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).

3, 2310, 2730, 30030, 39270, 43890, 46410, 51870, 53130, 60060, 62790, 66990, 67830, 71610, 72930, 78540, 79170, 81510, 82110, 84630, 85470, 87780, 90090, 91770, 92820, 94710, 98670, 99330, 101010, 102102, 103530, 103740, 106260, 106590, 108570, 110670, 111930

Offset

1,1

Comments

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).

Links

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.

Mathematica

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

Crossrefs

Cf. A001221, A239383, A336909.

Sequence in context: A286978 A293086 A293099 * A286715 A365706 A081176

Adjacent sequences: A336907 A336908 A336909 * A336911 A336912 A336913

Keyword

nonn

Author

Amiram Eldar, Aug 07 2020

Status

approved