64 Jean-Marie De Koninck

• the smallest number which is not the sum of five non zero distinct squares

(R.K. Guy [101], C20).

246

• the third number n such that σ2(n) is a perfect square: we indeed have

σ2(246) =

2902

(see the number 42).

248

• the smallest solution of σ(n) = σ(n + 37); the sequence of numbers satisfying

this equation begins as follows: 248, 302, 345, 518, 1142, 1298, 2108, 2174,

2505, 2678, 4604, . . .

249

• the ninth number n such that n ·

2n

− 1 is prime (see the number 115).

250

• the fourth solution of φ(n) = γ(n)2 (see the number 108).

251

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

distinct ways: 251 =

13

+

53

+

53

=

23

+

33

+

63

(see the number 1 009);

• the third positive solution x of the diophantine equation

x2

+ 999 =

y3

(R.

Steiner): the only positive solutions (x, y) of this diophantine equation are

(1, 10), (27, 12), (251, 40), (1 782, 147), (2 295, 174) and (3 370 501, 22 480).

255

• the number of Carmichael numbers

108

(Poulet, 1938); see the number 646;

• the seventh solution of φ(n) = φ(n + 1) (see the number 15).

256

• the largest known power of 2 which can be written as a sum of distinct powers

of 3: here 256 =

28

=

35

+

32

+ 3 + 1 (see R.K. Guy [101], B33).

257

• the fourth Fermat number: 257 =

223

+ 1; the first three are 3, 5 and 17;

• the smallest number n such that φ8(n) = 2, where φ8(n) stands for the eighth

iteration of the φ function (see the number 137).