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 =

· 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 = · q · p, where p = 2α+1 1 is a Mersenne prime = q.
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