336 Jean-Marie De Koninck

852 133 201

• the smallest prime factor of the Mersenne number 2157 − 1, whose complete

factorization is given by

2157

− 1 = 852133201 · 60726444167 · 1654058017289 · 2134387368610417;

it is the smallest Mersenne number with four prime factors (see the number

223).

865 950 624

• the tenth number n such that φ(n) + σ(n) = 4n (see the number 23 760).

880 346 227

• the smallest number n such that λ0(n) = λ0(n + 1) = . . . = λ0(n + 28) = 1,

where λ0 stands for the Liouville function; moreover, for this n, we also have

λ0(n + 29) = λ0(n + 30) = 1 (see the number 1 934).

893 648 277 (=

193

· 130 303)

• the fourth 18-hyperperfect number (see the number 1 333).

904 683 264

• the

23rd

number n such that φ(n) + σ(n) = 3n (see the number 312).

906 150 256

• the tenth number x such that

∑

n≤x

λ0(n) = 0 (M. Tanaka [192]), where λ0

stands for the Liouville function.

906 150 257

• the smallest number x 1 such that

∑

n

λ0(n) 0, where λ0 stands for the

Liouville function (P´ olya believed that

∑≤x

n≤x

λ0(n) ≤ 0 for all numbers x 1):

here

∑

n≤906150257

λ0(n) = 1 (see R.S. Lehman [123]).

912 985 153

• the largest number equal to the sum of the ninth powers of its digits (see the

number 146 511 208).