26 Jean-Marie De Koninck

81

• the only number n 1 whose sum of digits is equal to

√

n;

• the smallest number which can be written as the sum of three cubes and as the

sum of four cubes: 81 =

33

+

33

+

33

=

13

+

23

+

23

+

43.

82

• the smallest number which can be written as the sum of four cubes as well as

the sum of five cubes: here 82 =

13

+

33

+

33

+

33

=

13

+

13

+

23

+

23

+

43.

83

• the smallest number n which allows the sum

i≤n

1

i

to exceed 5: John V. Baxley

[19] developed a clever method to compute this number n; if nk stands for

the smallest number n which allows the sum

i≤n

1

i

to exceed k, we have the

following table:

k nk

2 4

3 11

4 31

5 83

6 227

7 616

8 1674

k nk

9 4550

10 12367

11 33617

12 91380

13 248397

14 675214

15 1835421

k nk

16 4989191

17 13562027

18 36865412

19 100210581

20 272400600

21 740461601

22 2012783315

• the smallest number n which requires six iterations of the Euler φ function

to reach the number 2; in other words, n = 83 is the smallest number such

that φ6(n) = 2, where φk stands for the kth iteration of φ; the sequence of the

smallest numbers n = n(k), k = 1, 2, . . ., requiring k iterations of the φ function

in order to reach the number 2 begins as follows: 3, 5, 11, 17, 41, 83, 137, 257,

641, 1097, 2329, 4369, 10537, 17477, . . . (see R.K. Guy [101], B41);

• the number appearing (as an exponent) in the diophantine equation

x83

−

y4871

= 1, an equation which cannot be solved by applying the “Inken criteria”,

the reason being that

834870

≡ 1 (mod

48712)

and

48712

≡ 1 (mod

832),

an

obstacle which was bypassed by M. Mignotte in 1992.

84

• the smallest solution of

σ(n)

n

=

8

3

(see the number 1488);