200 Jean-Marie De Koninck

where Pν stands for a ν digit prime number, while Cν is a ν digit composite

number158.

19 937

• the exponent of the 24th Mersenne prime 219 937 − 1 (Tuckerman, 1971).

19 999

• the largest number n such that f4(n) n, where f4(n) = f4([d1, d2, . . . , dr]) =

d1 4 + d2 4 + . . . + dr 4, where d1, d2, . . . , dr stand for the digits of n.

20 154

• the smallest number n such that σ(n), σ(n +1), σ(n +2), σ(n +3) and σ(n + 4)

have the same prime factors, namely here 2, 3, 5 and 7: the sequence of num-

bers satisfying this property begins as follows: 20 154, 29 395, 214 195, 764 392,

768 594, . . . (see the number 3 777).

20 160

• the

23rd

highly composite number (see the number 180).

20 161

• the smallest number n having the property that each number n can be

written as the sum of two abundant numbers (C.S. Ogilvy and J.T. Anderson

[155]).

20 201

• the smallest prime number equally distant, by a distance of 18, from the

preceding and following prime numbers: p2284 = 20 183, p2285 = 20 201 and

p2286 = 20 219.

20 440

• the eighth number n such that σ(φ(n)) = n (see the number 744).

20 496

• the only solution n

108

of σ(n) = 3n + 16.

158The

largest known prime of the form

10n

+ 1 is 101; one easily proves that if

10n

+ 1 is prime, n

must be a power of 2; but since one can easily check, using a computer, that

102k

+ 1 is composite

for 2 ≤ k ≤ 16, it follows that 10n + 1 is composite for 3 ≤ n ≤ 131 071 = 217 − 1.