144 Jean-Marie De Koninck

2 208

• the

11th

Keith number (see the number 197).

2 210

• the smallest number n which can be written as the sum of the squares of two

prime numbers in three distinct ways: 2 210 =

192 +432

=

232 +412

=

292 +372

(see the number 338);

• the smallest number n such that ω(n) + ω(n + 1) + ω(n + 2) = 10: here

2 210 = 2 · 5 · 13 · 17, 2 211 = 3 · 11 · 67 and 2 212 = 22 · 7 · 79; if nk stands for

the smallest number n such that ω(n) + ω(n + 1) + ω(n + 2) = k, we have the

following table:

k nk

4 4

5 10

6 20

7 68

8 154

k nk

9 644

10 2 210

11 6 578

12 35 308

13 92 378

k nk

14 310 154

15 1 042 404

16 5 617 870

17 35 515 634

18 184 055 430

2 281

• the exponent of the

17th

Mersenne prime

22 281

− 1 (Robinson, 1952).

2 285

• the fifth number which is not a palindrome, but whose square is a palindrome

(see the number 26).

2 295

• the fifth positive solution x of the diophantine equation

x2

+ 999 =

y3

(see the

number 251).

2 303

• the smallest solution of τ (n+1)−τ (n) = 21; the sequence of numbers satisfying

this equation begins as follows: 2 303, 24 649, 67 599, 85 849, 104 975, 132 495,

283 023, . . . (see the number 399).

2 309

• the third prime number p of the form p =

r

i=1

pi − 1: 2 309 = 2 · 3 · 5 · 7 · 11 − 1

(see the number 317).