188 Jean-Marie De Koninck

12 168 (= 23 · 32 · 132)

• the second powerful number n such that n − 1 is a cube, namely

233;

the

smallest number n satisfying this property is 9, and there are no others

1015.

12 213

• the largest known number n such that

2n

− n is prime (see the number 261).

12 251

• the

34th

Lucas prime number (see the number 613).

12 367

• the smallest number n which allows the sum

i≤n

1

i

to exceed 10 (see the number

83).

12 379

• the

19th

number n such that n ·

2n

− 1 is prime: in fact, it is the third prime

number n satisfying this property, the second being 751, and the first being 3

(see the number 115).

12 546

• the number of integer zeros of the function M(x) :=

n≤x

µ(n) in the interval

[1,

107]

(see the number 92).

12 558 (= 2 · 3 · 7 · 13 · 23)

• the eighth ideal number (see the number 390).

12 719

• the ninth Lucas-Carmichael number (see the number 399).

12 721

• the smallest number n such that P (n) P (n + 1) . . . P (n + 6): for

this number n, we also have P (n) P (n + 1) . . . P (n + 7), that is

12721 6361 4241 3181 509 101 89 43 (see the number 1 851).