180 Jean-Marie De Koninck
8 925
the smallest solution of σ(n) = 2n + 6; the sequence of numbers satisfying this
property begins as follows: 8925, 32 445, 442 365, . . .
151
8 970 (= 2 · 3 · 5 · 13 · 23)
the seventh ideal number (see the number 390).
9 011
the
11th
prime number p such that
(3p
1)/2 is itself a prime number (see the
number 1 091).
9 091
one of the only two prime numbers p (the other is 11) with the property that
any number of the form abcdeabcde is
divisible152
by p.
9 272
the seventh bizarre number (see the number 70).
9 352
the smallest number n such that β(n) β(n + 1) . . . β(n + 6): here
176 246 1564 1876 2341 3122 4681 (see the number 714;
compare with the number 46 189).
9 374
the seventh solution of σ(φ(n)) = σ(n) (see the number 87).
9 376
the only four digit automorphic number: 9
3762
= 87 909 376 (see the number
76).
151It
is easy to prove that n =

· p, with p a prime and α a positive integer, is a solution of
σ(n) = 2n + 6 if and only if p =
2α+1
7: we thus find that the smallest solution of the form
n =
2αp
is obtained when α = 38 and p =
2α+1
7 = 549 755 813 881, which yields the solution
n = 151 115 727 449 904 501 489 664; this phenomenon repeats itself when α = 714 and not before.
152The
proof of this result is trivial. Indeed, for a number n of the form abcdeabcde, we have
n = a ·
109
+ b ·
108
+ c ·
107
+ d ·
106
+ e ·
105
+ a ·
104
+ b ·
103
+ c ·
102
+ d · 10 + e
= a ·
104(105
+ 1) + b ·
103(105
+ 1) + c ·
102(105
+ 1) + d ·
10(105
+ 1) +
e(105
+ 1),
and since 105 + 1 = 11 · 9091, the result follows.
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