Those Fascinating Numbers 161

4 320 (= 25 · 33 · 5)

• the smallest solution of

σ(n)

n

=

7

2

; the sequence of numbers satisfying this pro-

perty begins as follows: 4 320, 4 680, 26 208, 20 427 264, 197 064 960, . . .

4 329

• the third number whose square can be written as the sum of three fourth

powers: 4

3292

=

364

+

454

+

604

(see the number 481).

4 355

• the smallest solution of τ (n+1)−τ (n) = 19; the sequence of numbers satisfying

this property begins as follows: 4355, 4899, 6083, 12099, 12543, 12995, 16899,

. . . (see the number 399).

4 364

• the ninth solution of σ(n) = σ(n + 1) (see the number 206).

4 369

• the smallest number n such that

φ12(n)

= 2, where

φ12(n)

stands for the

12th

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

4 374

• the second number n such that P (n)

4

√

n and P (n + 1)

4

√

n + 1: here

P (4374) =√ P (2 ·

37)

= 3 8.13 . . . = 4

√

4374 and P (4375) = P

(54

· 7) = 7

8.13 . . . = 4 4375 (see the number 2 400).

4 418

• the smallest solution of σ(n + 13) = σ(n) + 13.

4 423

• the exponent of the

20th

Mersenne prime

24 423

− 1 (Hurwitz, 1961).

4 471

• the fourth odd number k such that

2n

+ k is composite for all n k (see the

number 773): in fact,

2n

+ 4 471 is composite for all n 33 548 and prime for

n = 33 548.