130 Jean-Marie De Koninck

1 469

• the second number n such that φ(n) = 4φ(n + 1) (see the number 629).

1 477

• the

17th

number n such that n! + 1 is prime (see the number 116).

1 481

• the first member p of the third 6-tuple (p, p +2, p +6, p +8, p +12, p +18) made

up entirely of prime numbers: the smallest 6-tuple satisfying this property is

(5,7,11,13,17,23) and the second is (11, 13, 17, 19, 23, 29).

1 484

• the smallest number n such that φ(n) φ(n + 1) φ(n + 2) φ(n + 3): here

624 720 742 1486 (see the number 105).

1 487

• the smallest prime factor of the Mersenne number

2743

− 1.

1 488

• the third solution of

σ(n)

n

=

8

3

; the sequence of numbers satisfying this equation

begins as follows: 84, 270, 1 488, 1 638, 24 384, 100 651 008, . . . 130

1 491

• the smallest number n such that the decimal expansion of

2n

contains five

consecutive zeros (see the number 53).

1 492

• the smallest number n such that the decimal expansion of

2n

contains six

consecutive zeros (see the number 53).

130It

is easy to prove that if n =

2α

· 3 · p, where p =

2α+1

− 1 is a Mersenne prime 3 (that is if

n = 3m, where m is an even perfect number 6), then n is a solution of

σ(n)

n

=

8

3

; this guarantees

the existence of at least 45 solutions of this form, the first six being 84, 1 488, 24 384, 100 651 008,

25 769 607 168 and 412 316 073 984. In fact, the only solutions n 5 ·

108

which are not of this form

are 270 and 1 638. Moreover, one can prove that equation

σ(n)

n

=

2q+2

q

, with q an odd prime, has

the solutions n = 2α · q · p, where p = 2α+1 − 1 is a Mersenne prime = q.