50 Jean-Marie De Koninck

168

• the smallest number m such that equation σ(x) = m has exactly six solutions,

namely 60, 78, 92, 123, 143 and 167;

• the number of prime numbers 1000;

• the largest known number k such that the decimal expansion of 2k does not

contain the digit 2; indeed, using a computer, one can verify that

2168

= 374144419156711147060143317175368453031918731001856

does not contain the digit 2, while each number

2k,

for k = 169, 170, . . . , 3000,

contains it (see the number 71);

• the smallest solution of σ2(n) = σ2(n + 14); the sequence of numbers satisfying

this equation begins as follows: 108, 785, 7553, 6632633, 9535673, . . .

169

• the smallest perfect square m3 2 for which there exist numbers m1 and m2 such

that mi 2 − (mi − 1)2 = mi−1 2 for i = 2, 3: here 169 = 132 = 122 + 52 =

122 + 42 + 32.

170

• the smallest number n such that λ0(n) = λ0(n + 1) = . . . = λ0(n + 6) = −1,

where λ0 stands for the Liouville function;

• the smallest number n 1 such that φ(n)σ(n)

is63

a fourth power: here

φ(n)σ(n) =

124;

the sequence of numbers satisfying this property begins as

follows: 1, 170, 595, 714, 121056, 480441, 529620, 706063, 706237, 729752,

755972, 815654, . . .

171

• the largest known number n such that the binomial coeﬃcient

(2n)

n

is not

divisible by the square of an odd prime number (R.K. Guy [101], B33): here

342

171

=

25

· 3 · 5 · 11 · 17 · 29 · 31 · 37 · 43 · 47 · 59 · 61 · 67 · 89 · 97 · 101

·103 · 107 · 109 · 113 · 173 · 179 · 181 · 191 · 193 · 197 · 199

·211 · 223 · 227 · 229 · 233 · 239 · 241 · 251 · 257 · 263 · 269

·271 · 277 · 281 · 283 · 293 · 307 · 311 · 313 · 317 · 331 · 337.

63One

can prove that there exist infinitely many numbers n such that φ(n)σ(n) is a perfect square.

In fact, Florian Luca can prove (private communication) that, for each positive integer s for which

there exist numbers n0 and x0 such that φ(n0)σ(n0) =

sx2,

0

equation φ(n)σ(n) =

sx2

has infinitely

many solutions (n, x).