52 Jean-Marie De Koninck

k nk nk

4 8 8

5 13 13

6 22 22

7 38 37

8 63 63

9 105 105

10 177 177

11 296 296

12 495 495

k nk nk

13 828 828

14 1386 1386

15 2318 2318

16 3879 3878

17 6489 6488

18 10854 10854

19 18158 18157

20 30375 30374

21 50811 50811

k nk nk

22 84998 84998

23 142187 142187

24 237853 237853

25 397885 397885

26 665589 665589

27 1113411 1113410

28 1862534 1862534

29 3115683 3115683

30 5211973 5211973

178

• the eighth number n for which the distance from

en

to the nearest integer is the

smallest, that is for which κ(i) κ(n) for all i n, where κ(n) := min

m≥1

|en

− m|:

the sequence of these numbers n begins as follows: 1, 3, 8, 19, 45, 75, 135, 178,

209, 732, 1351, 1907, . . .

180

• the

11th

highly composite number: a number n is said to be highly composite

if τ (n) τ (m) for all numbers m n: here τ (180) = 18; the sequence of

numbers satisfying this property begins as follows: 1, 2, 4, 6, 12, 24, 36, 48, 60,

120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160,

25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200,

332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880,

3603600, 4324320, 6486480, 7207200, 8648640, . . . (see J.L. Nicolas [150]);

• the only solution n

109

of σ(n) = 3n + 6;

• the smallest number n 1 such that σ(n) and σ2(n) have the same prime

factors, that is such that γ(σ(n)) = γ(σ2(n)): here the common factors are 2,

3, 7 and 13; the sequence of numbers satisfying this property begins as follows:

180, 1444, 12996, 23805, 36100, 52020, 60228, 64980, 68832, 95220, 301140,

324900, 344160, 481824, . . .

181

• the largest integer solution x of equation

x2

+ 7 =

2n:

in 1960, Nagell (see

L.J. Mordell [144]) established that this diophantine equation has only five so-

lutions, namely (x, n) = (1, 3), (3,4), (5,5), (11,7) and (181,15).

182

• the smallest solution of σ(n) = σ(n + 13); the sequence of numbers satisfying

this equation begins as follows: 182, 782, 1965, 2486, 2678, 2685, 12141, 12441,

17342, 21242, . . .