56 Jean-Marie De Koninck

k p0(k) rank

1 2 1

2 3 2

3 5 and 7 3 and 4

4 13 6

5 23 9

6 47 15

k p0(k) rank

7 113 30

8 199 46

9 283 61

10 467 91

11 887 154

12 1 627 258

k p0(k) rank

13 2 083 409

14 4 297 590

15 6 397 834

16 10 343 1 270

17 16 111 1 876

18 24 251 2 699

200

• the only solution n 1012 of σ(n) = 2n + 65 (see the number 196);

• the only number n = 4 and smaller than 1010 such that n divides σ(n) + σ2(n).

201

• the smallest composite number of the form k · 10k + 1; the sequence of num-

bers satisfying this property begins as follows: 201, 40 001, 500 001, 6 000 001,

70 000 001, 800 000 001, 100 000 000 001, 1 100 000 000 001, . . . (see the number

363).

203

• the sixth Bell number (see the number 52).

204

• the total number of squares one can reproduce on a check board, namely 1 of

dimension 8 × 8, 4 of dimension 7 × 7, 9 of dimension 6 × 6, 16 of dimension

5 × 5, 25 of dimension 4 × 4, 36 of dimension 3 × 3, 49 of dimension 2 × 2, and

finally 64 of dimension 1 × 1;

• the second number (and possibly the largest !) whose square is the sum of

three consecutive cubes

(2042

=

233

+

243

+

253);

the smallest one is 6

(62

=

13

+

23

+

33);

• the third solution w of the aligned houses problem (see the number 35).

205

• the number of twin prime pairs 104 (see the number 1 224).

206

• the second number n such that σ(n) = σ(n + 1) (see the number 14, as well as

the numbers 1 253 and 1 919).