94 Jean-Marie De Koninck

567

• the smallest number n such that if A and B stand respectively for the set

of digits of n and of n2, then A ∪ B = {1, 2, 3, . . . , 9} and A ∩ B = ∅: here

5672 = 321 489; the only other number with this property is 854.

571

• the smallest number n which allows the sum

m≤n

1

σ(m)

to exceed 5 (see the

number 129).

575

• the smallest solution of τ (n + 1) − τ (n) = 15 (see the number 399).

576

• the smallest number m such that equation σ(x) = m has exactly 11 solutions,

namely 210, 282, 310, 322, 345, 357, 382, 385, 497, 517 and 527.

577

• the sixth number n such that

n2

− 1 is powerful: here

5772

− 1 =

27

·

32

·

172

(see the number 485).

581

• the second Canada perfect number: 52 +82 +12 = 7+83 (see the number 125).

582

• the only number 1 which is equal to the sum of the squares of the factorials

of its digits in base 6: here 582 = [2, 4, 1, 0]6 = 2!2 + 4!2 + 1!2 + 0!2 (see the

number 145).

584

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

586

• the ninth number x such that

∑

n≤x

λ0(n) = 0, where λ0 stands for the Liouville

function: the sequence of numbers satisfying this property begins as follows: 2,

4, 6, 10, 16, 26, 40, 96, 586, 906 150 256, . . . (see the number 906 150 256).