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

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1

http://dx.doi.org/10.1090/mbk/064/01