20 Jean-Marie De Koninck

58

• the second Smith number (see the number 22): here 58 = 2 · 29 and 5 + 8 =

13 = 2 + 2 + 9;

• the fourth number n such that the distance from πn to the nearest integer is

the smallest: the sequence of these numbers n begins as follows: 1, 2, 3, 58, 81,

157, 1030,. . .

59

• the second irregular prime number (the smallest is 37): a prime number is said

to be irregular if it is not regular; a prime number p is said to be regular if p

does not divide the numerator of B2k for 2k = 2, 4, . . . , p − 3, where Bi is the

ith Bernoulli number; one can also prove that a prime number p is regular if

and only if p2 does not divide any of the numbers 1k + 2k + . . . + (p − 1)k for

k = 2, 4, . . . , p − 3; the irregular prime numbers smaller than 1 000 are 37, 59,

67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353,

379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587,

593, 607, 613, 617, 619, 631, 647, 653, 659, 673, 677, 683, 691, 727, 751, 757,

761, 773, 797, 809, 811, 821, 827, 839, 877, 881, 887, 929, 953 and 971; in 1915,

Jensen proved that there exist infinitely many irregular primes; but it is still

not known if there exist infinitely many regular primes.

60

• one of the five known unitary perfect numbers (see the number 6);

• the smallest dihedral 3-perfect number (see the number 5 472);

• the smallest number n such that

Ω(n)ω(n)

n; the sequence of numbers satis-

fying this property begins as follows: 60, 120, 210, 420, 840, 1260, 1680, 2310,

2730, 3360, 4620, 5460, 6930, 7140, 9240, . . . (see the number 3 569 485 920 for

more on the behavior of the quotient

Ω(n)ω(n)

n

).

61

• the exponent of the ninth Mersenne prime

261

− 1 (Pervouchine, 1883; and

Seelhoff, 1886);

• the rank of the prime number which appears the most often as the ninth prime

factor of an integer : p61 = 283 (see the number 199);

• the smallest number n such that n

pn

log pn − 1

(see the number 492);28

28It

is interesting to mention that Ramanujan often used the approximation π(x) ≈

x

log x − 1

.