148 Jean-Marie De Koninck

2 520

• the largest of the six highly composite numbers (the other five are 1, 2, 6, 12

and 60) which divide each of the larger highly composite numbers (S. Ratering

[167]);

• the value of [1, 2, 3, . . . , 10], that is the least common multiple of the numbers

1 to 10;

• the 18th highly composite number (see the number 180).

2 522

• the smallest number n such that Ω(n) = Ω(n + 1) = . . . = Ω(n + 5): here this

common value is 3 (see the number 602).

2 525

• the smallest number n which allows the sum

m≤n

1

σ(m)

to exceed 6 (see the

number 129).

2 548

• the second number which is not a cube and which can be written as the sum

of the cubes of its prime factors: 2 548 =

22

·

72

· 13 =

23

+

73

+

133

(see the

number 378).

2 580

• the 12th Keith number (see the number 197).

2 592

• the only number of the form abca such that abca =

ab

·

ca;

indeed, here we have

2 592 =

25

·

92.

2 593

• the first term of the smallest sequence of five consecutive prime numbers all

of the form 4n + 1; denoting by qk the first term of the first sequence of k

consecutive prime numbers all of the form 4n + 1, we have the following table:

k qk

1 5

2 13

3 89

4 389

5 2 593

k qk

6 11 593

7 11 593

8 11 593

9 11 593

10 373 649

k qk

11 766 261

12 3 358 169

13 12 204 889

14 12 270 077

15 12 270 077

k qk

16 12 270 077

17 297 387 757

18 297 779 117

19 297 779 117

20 1 113 443 017

(see the number 463 for the analogue problem with 4n + 3).