328 Jean-Marie De Koninck

299 980 800 (= 211 · 33 · 52 · 7 · 31)

• the third solution of

σ(n)

n

=

13

3

(see the number 18 506 880).

305 635 357 (= 43 · 1039 · 6841)

• the smallest composite number n such that σ(n + 4) = σ(n) + 4: indeed, since

305 635 357 = 43 · 1039 · 6841, we have

σ(305635357) = σ(43) · σ(1039) · σ(6841) = 44 · 1040 · 6842 = 313089920

and since 305635361 = 41 · 7454521, we have

σ(305635361) = σ(41) · σ(7454521) = 42 · 7454522 = 313089924;

hence, σ(305635357 + 4) = σ(305635357) + 4 (see also the number 434).

315 746 063

• the third prime number p such that

83p−1

≡ 1 (mod

p2)

(see the number

4 871).

321 197 185 (= 5 · 19 · 23 · 29 · 37 · 137)

• the smallest Carmichael number which is the product of six prime numbers (see

the number 41 041).

333 333 331 (= 17 · 19 607 843)

• the smallest composite number of the form

1

3

(10k

− 7): here k = 9 (an observa-

tion due to A. Makowski); the numbers k ≤ 4 000 for which

1

3

(10k

− 7) is prime

are k = 2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, 784, 1 732 and

1 918.

341 118 307

• the first term of the smallest sequence of 18 consecutive prime numbers all of

the form 4n + 3 (see the number 463).

348 364 800

• the fourth number which is equal to the product of the factorials of its digits

in base 8: 348 364 000 = [2, 4, 6, 0, 7, 2, 0, 0, 0, 0]8 (see the number 17 280).