146 Jean-Marie De Koninck

2 378

• the fifth solution y of the Fermat-Pell equation x2 − 2y2 = 1: here (x, y) =

(3363, 2378) (see the number 99).

2 383

• the smallest prime factor of the Mersenne number

2397

− 1, whose complete

factorization is given by

2397

− 1 = 2383 · 6353 · 50023 · 53993 · 202471 · 5877983

·814132872808522587940886856743

·1234904213576000272542841146073

·6597485910270326519900042655193;

it is the smallest Mersenne number with exactly nine prime factors (see the

number 223 for the list of the smallest Mersenne numbers with a given number

of prime factors).

2 400

• the smallest number n such that P (n) ≤

4

√

n and P (n + 1) ≤

4

√

n + 1: here

P (2400) = P (25 · 3 · 52) = 5 6.99 . . . =

4

√

2400 and P (2401) = P (74) =

7 =

4

√

2401; the sequence of numbers satisfying this property begins as fol-

lows: 2400, 4374, 123200, 165375, 194480, 228095, 282624, 328509, 336140,

601425, . . . : one can prove that this sequence is

infinite139;

if nk stands for the

smallest number n such that max(P (n), P (n + 1)) ≤ (n +

1)1/k,

then n2 = 8,

n3 = n4 = 2 400, n5 = 5 909 560 and n6 = 1 611 308 699.

2 401

(=74)

• the smallest number n 1 whose sum of digits is equal to

4

√

n: the others are

234 256, 390 625, 614 656 and 1 679 616;

• the third perfect square which is also a Smith number (see the number 22):

2401 = 74 and 2 + 4 + 1 = 7; the smallest two are 361 and 1 600.

2 418

• the smallest number n such that

Li2(x)

ex

log x

Li

x

e

for all x ≥ n (see B.C.

Berndt [21] as well as the number 38 358 837 677);

• the seventh number n such that σ(φ(n)) = n (see the number 744).

139Indeed, A. Balog & Z. Ruzsa [14] have proved that, for each ε 0, the set {n : max(P (n), P (n+

1))

nε}

is of positive density, implying in particular that the set of numbers n such that P (n) ≤

4

√

n and P (n + 1) ≤ 4

√

n + 1 is of positive density.