302 Jean-Marie De Koninck

17 907 119

• the smallest number n satisfying φ(n) = 5φ(n + 1); the sequence of num-

bers satisfying this property begins as follows: 17907119, 18828809, 31692569,

73421039, 179467469, . . . (see the number 629).

18 003 384 (= 23 · 38 · 73)

• the fifth number n having at least two distinct prime factors and such that

β(n)3|B1(n): here (2 + 3 + 7)3|(23 + 38 + 73) (see the number 5 120).

18 035 622

• the smallest of the first four consecutive numbers being each divisible by a

cube 1: 18 035 622 = 2 ·

34

· 11 · 29 · 349, 18 035 623 =

173

· 3671, 18 035 624 =

23

· 163 · 13831, 18 035 625 = 3 ·

54

· 9619 and 18 035 626 = 2 ·

73

· 61 · 431 (see

the number 844).

18 506 880 (=

27

·

35

· 5 · 7 · 17)

• the smallest solution of

σ(n)

n

=

13

3

: the only solutions n 109 of this equation

are 18 506 880, 36 197 280 and 299 980 800.

18 673 201

• the second number n such that φ(n + 1) = 5φ(n) (see the number 11 242 770).

19 099 919

• the smallest prime number q1 such that each number qi = 2qi−1 + 1 is prime

for i = 2, 3, . . . , 8: such a sequence of prime numbers is called a Cunningham

chain (see the number 1 122 659).

19 592 147

• the fourth number which can be written in two distinct ways as the sum of

two co-prime numbers each with an index of composition ≥ 5: 19 592 147 =

313

+

213

·

133

=

24

·

38

+

117,

these last four numbers having as index of

composition 13, 5.12746, 6.45259 and 7 respectively (see the number 371 549).

19 720 007

• the ninth solution of σ2(n) = σ2(n + 2) (see the number 1 079).