Unusual number

In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than \sqrt{n} (sequence A064052 in OEIS). All prime numbers are unusual.

A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non-\sqrt{n}-smooth.

If we denote the number of unusual numbers less than or equal to n by u(n) then u(n) behaves as follows:

n u(n) u(n) / n
10 6 0.6
100 67 0.67
1000 715 0.715
10000 7319 0.7319
100000 70128 0.70128

Richard Schroeppel proved in 1972 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words:

\lim_{n \rightarrow \infty} \frac{u(n)}{n} = \ln(2) = 0.693147 \dots\, .

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