Those Fascinating Numbers 261

465 125 (= 53 · 612)

• the smallest powerful number n such that n − 1 is a perfect square (namely

6822):

the second number satisfying this property is

1 610 006 506 595 061 125 = 1 268 860

3182

+ 1 =

53

·

172

·

612

·

1094412

(see the number 682).

467 458

• the third even number n such that σI (n) = σI (n + 2) (see the number 54 178).

470 449

• the

13th

number n such that

n2

− 1 is powerful: here

470

4492

− 1 =

25

·

35

·

52

·

112

·

972

(see the number 485).

470 832

• the eighth solution y of the Fermat-Pell equation x2 − 2y2 = 1: here (x, y) =

(665857, 470832); see the number 99.

472 601

• the second composite number n such that σ(n+56) = σ(n)+56 (see the number

76 571).

480 441

• the sixth number n such that φ(n)σ(n) is a fourth power: here

φ(480441)σ(480441) = 6724 (see the number 170).

480 852

• the smallest number n such that π(n) = n/12 (see the number 330).

481 824

• the

13th

number n such that σ(n) and σ2(n) have the same prime factors,

namely the primes 2, 3, 5, 7 and 13 (see the number 180).