314 Jean-Marie De Koninck

49 799 889

• the smallest number n such that Ω(n) = Ω(n + 1) = . . . = Ω(n + 9): here this

common value is 4 (see the number 602).

49 989 744

• the smallest Niven number which is followed by at least 100 numbers (in fact

here by 107) which are not Niven numbers.

50 847 534

• the number of prime numbers 109.

51 767 910

• the smallest number n such that

min(λ(n), λ(n + 1), λ(n + 2), λ(n + 3), λ(n + 4), λ(n + 5))

6

5

:

here

min(λ(n), λ(n + 1), λ(n + 2), λ(n + 3), λ(n + 4), λ(n + 5))

≈ min(1.22783, 1.40609, 1.23092, 1.62917, 1.36986, 1.27675) = 1.22783;

the next number satisfying the above inequality is n = 25 479 451 773; most

likely, there exist infinitely many numbers n satisfying the above inequality

(see the number 14 018 750, as well as J.M. De Koninck & N. Doyon [48]).

52 021 242

• the 18th number n such that φ(n) + σ(n) = 3n (see the number 312).

52 521 875 (=

55

·

75)

• the second number n 1 whose sum of digits is equal to

5

√

n (see the number

17 210 368).

53 262 468

• the seventh number n such that σ3(n) is a perfect square: here σ3(53262468) =

4236745968002

(see the number 345).

56 598 313

• the largest known prime number p such that

10p−1

≡ 1 (mod

p2)

(see the

number 487).