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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).
2
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Aug 07 2020
STATUS
approved