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 =

2α

· 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.