A319923 - OEIS

%I #19 Dec 14 2024 09:14:03

%S 121,133,143,161,169,187,203,209,217,221,247,253,259,287,289,299,301,

%T 319,323,329,341,343,361,371,377,391,403,407,413,427,437,451,469,473,

%U 481,493,497,511,517,527,529,533,539,551,553,559,581,583,589,611,623,629,637,649,667,671

%N Quasi-primes: composite numbers k such that the least prime factor is greater than k^(1/(log log k)^2).

%C Obviously the least prime factor of a composite number k can't be greater than sqrt(k). However, it can be greater than some function that grows more slowly than sqrt(k); k^(1/(log log k)^2) is just one possibility.

%H Encyclopedia of Mathematics, <a href="http://www.encyclopediaofmath.org/index.php?title=Quasi-prime_number">Quasi-prime number</a>

%e 539 = 7^2 * 11, and 539^(1/(log log 539)^2) = 6.42358133145183..., which is less than least prime factor 7. Hence 539 is in the sequence.

%e 540 = 2^2 * 3^3 * 5, and 540^(1/(log log 540)^2) = 6.423273267433689..., which is greater than least prime factor 2, so 540 is not in the sequence.

%t Select[Range[1000], Not[PrimeQ[#]] && FactorInteger[#][[1, 1]] > #^(1/Log[Log[#]]^2) &]

%Y Cf. A251728.

%K nonn

%O 1,1

%A _Alonso del Arte_, Oct 01 2018