4 Jean-Marie De Koninck

11

• the smallest prime number p such that 3p−1 ≡ 1 (mod p2): the only other

prime number p 232 satisfying this congruence is p = 1 006 003 (see Riben-

boim [169], p. 347)3;

• the smallest number n which allows the sum

i≤n

1

i

to exceed 3 (see the number

83).

12

• the smallest pseudo-perfect number: we say that a number is pseudo-perfect if

it can be written as the sum of some of its proper divisors: here 12 = 6 + 4 + 2;

in 1976, Erd˝ os proved that the set of pseudo-perfect numbers is of positive

density (see R.K. Guy [101], B2);

• the smallest number m for which equation σ(x) = m has exactly two solutions,

namely 6 and 11;

• the only number n 1 such that σ(γ(n)) = n;

• the smallest sublime number: we say that a number n is sublime if τ (n) and

σ(n) are both perfect numbers: here τ (12) = 6 and σ(12) = 28; this concept was

introduced by Kevin Ford; the only other known sublime number is

2126(261

−

1)(231

−

1)(219

−

1)(27

−

1)(25

−

1)(23

− 1).

13

• the second Wilson prime (see the number 5);

• the prime number which appears the most often as the fourth prime factor of

an integer, namely 31 times in 5005 (see the number 199);

• the smallest prime number p such that

23p−1

≡ 1 (mod

p2):

the only prime

numbers p

232

satisfying this congruence are 13, 2 481 757 and 13 703 077 (see

Ribenboim [169], p. 347);

• the third horse number: we say that n is a horse number if it represents the

number of possible results accounting for ties, in a race in which k horses

participate; thus, if Hk is the

kth

horse number, one can

prove4

that

Hk =

k

i=1

ik

⎛

⎝

k−i

j=0

(−1)j

j + i

j

⎞

⎠

;

the first 20 terms of the sequence (Hk)k≥1 are 1, 3, 13, 75, 541, 4683, 47293,

545835, 7087261, 102247563, 1622632573, 28091567595, 526858348381,

10641342970443, 230283190977853, 5315654681981355, 130370767029135901,

3385534663256845323, 92801587319328411133 and 2677687796244384203115.

3As is the case for the Wieferich primes (see the number 1 093), it is not known if this sequence

of numbers is infinite.

4A formula established by Charles Cassidy (Universit´ e Laval).