Those Fascinating Numbers 1 • the only number which divides all the others. 2 • the only even prime number. 3 • the prime number which appears the most often as the second prime factor of an integer, and actually with a frequency of 1 6 (see the number 199 for the list of those prime numbers which appear the most often as the kth prime factor of an integer, for any fixed k ≥ 1). • the smallest Mersenne prime (3 = 22 −1): a prime number is called a Mersenne prime if it is of the form 2p − 1, where p is prime (see the number 131 071 for the list of all Mersenne primes known as of May 2009) • the prime number which appears the most often as the second largest prime factor of an integer, that is approximately (1 + log 2 + 3 2 log 3)x/ log x times amongst the positive integers n ≤ x (see J.M. De Koninck [44]) • the smallest triangular number 1: a number n is said to be triangular if there exists a number k such that n = 1 + 2 + 3 + . . . + k = k(k + 1) 2 • • • • • • • • • • . . 1 http://dx.doi.org/10.1090/mbk/064/01

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