Those Fascinating Numbers 299

13 631 489

• the smallest prime factor of F18 =

2218

+ 1, whose partial factorization is given

by

F18 = 13631489 · 81274690703860512587777 · C78907.

13 703 077

• the third prime number p such that 23p−1 ≡ 1 (mod p2) (see the number 13).

13 821 503

• the smallest prime factor of the Mersenne number 2193 − 1, whose complete

factorization is given by

2193

− 1 = 13821503 · 61654440233248340616559 · 14732265321145317331353282383.

14 018 750

• the smallest number n such that

min(λ(n), λ(n + 1), λ(n + 2), λ(n + 3), λ(n + 4))

5

4

:

here

min(λ(n), λ(n + 1), λ(n + 2), λ(n + 3), λ(n + 4))

≈ (1.64261, 1.25044, 1.2668, 1.30976, 1.4742) = 1.25044;

the list of numbers satisfying the above inequality begins as follows: 14 018 750,

82 564 350, 3 387 574 069, 10 514 077 440, 11 807 429 372, 17 265 859 071,

21 036 065 792, . . . ; most likely there are infinitely many such numbers190.

14 459 929

• the largest number which can be written as the sum of the seventh powers of

its digits (see the number 1 741 725).

14 520 576 (= 28 · 3 · 7 · 37 · 73)

• the largest number n 109 such that

σ(n)

n

=

k

6

for some number k satisfying

(k, 6) = 1, here with k = 19 (see the number 22 932).

190Indeed,

J.M. De Koninck & N. Doyon [48] proved that, for each integer k ≥ 2, the probability

that there is only a finite number of numbers n such that

min (λ(n), λ(n + 1), . . . , λ(n + k − 1))

k

k − 1

is equal to zero.