184 Jean-Marie De Koninck

10 343

• the prime number which appears the most often as the

16th

prime factor of an

integer (see the number 199).

10 368

• the smallest number n = [d1, d2, . . . , dr] such that (d1 +r)·(d2 +r −1)·. . .·(dr +

1) = n; the set of numbers satisfying this property is finite and its members

are 10 368, 34 496, 9 317 179 872, 2 974 959 187 200 and 3 942 878 404 608 000;

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

in base 5: 10 368 = [3, 1, 2, 4, 3, 3]5 = 3! · 1! · 2! · 4! · 3! · 3! (see the number 144).

10 370

• the smallest number n which can be written as the sum of the squares of two

prime numbers in four distinct ways: 10 370 =

132

+

1012

=

312

+

972

=

592

+

832

=

712

+

732

(see the number 338).

10 430

• the eighth bizarre number (see the number 70).

10 529

• the smallest prime number p such that Ω(p − 1) = Ω(p + 1) = 7: here 10 528 =

25

· 7 · 47 and 10 530 = 2 ·

34

· 5 · 13 (see the number 271).

10 537

• the smallest number n such that

φ13(n)

= 2, where

φ13(n)

stands for the

13th

iteration of the φ function (see the number 137).

10 585

• the eighth Carmichael number (see the number 561).

10 604

• the

18th

solution of φ(n) = φ(n + 1) (see the number 15).

10 691

• the

33rd

Lucas prime number (see the number 613).