166 Jean-Marie De Koninck

5 187

• the

17th

solution of φ(n) = φ(n + 1) (see the number 15).

5 234

• the largest solution y of the diophantine equation x2 − 17 = y3: T. Nagel

[147] proved that this diophantine equation has exactly 16 solutions (x, y),

namely the solutions (±3, −2), (±4, −1), (±5, 2), (±9, 4), (±23, 8), (±282, 43),

(±375, 52), (±378661, 5234) (see also Sierpinski [185], p.104).

5 264

• the smallest number n such that n and n+1 each have six prime factors counting

their multiplicity: 5 264 =

24

· 7 · 47 and 5 265 =

34

· 5 · 13 (see the number 135).

5 312

• the 16th number n such that n · 2n − 1 is prime (see the number 115);

• the second number n such that Eφ(n) := φ(n + 1) − φ(n) satisfies Eφ(n + 1) =

Eφ(n): here the common value of Eφ is 16, since φ(5 312) = 2 624, φ(5 313) =

2 640 and φ(5 314) = 2 656; the sequence of numbers satisfying this property

begins as follows: 5 186, 5 312, 273 524, . . .

5 337

• the sixth composite number n such that σ(n + 6) = σ(n) + 6 (see the number

104).

5 346 (= 2 · 35 · 11)

• the fourth number n having at least two distinct prime factors and such that

B1(n) = β(n)2: here 2 + 35 + 11 = (2 + 3 + 11)2 (see the number 144).

5 355

• the eighth odd abundant number (see the number 945).

5 361

• the number of integer zeros of the function M(x) :=

n≤x

µ(n) located in the

interval [1,

106]

(see the number 92).