Those Fascinating Numbers 165

5 044 (= 22 · 13 · 97)

• the smallest number n such that n, n + 2, n + 4, . . . , n + 10 are all divisible by

a square 1.

5 101

• the fifth odd number k such that 2n + k is composite for all n k (see the

number 773): in fact, 2n + 5 101 is composite for each n ≤ 5 759 and prime for

n = 5 760.

5 104

• the smallest number which can be written as the sum of three distinct cubes

in three distinct ways:

5 104 =

13

+

123

+

153

=

23

+

103

+

163

=

93

+

103

+

153

(see the number 1 009).

5 105

• the largest number k such that inequality θ(pk) k(log k +log log k −1) (where

θ(x) =

∑

p≤x

log p) is false (G. Robin [175]).

5 120 (=

210

· 5)

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

β(n)3|B1(n):

here (2 +

5)3|(210

+ 5); the only numbers

108

satisfying this

property are 5 120, 419 904, 885 735, 5 315 625 and 18 003 384.

5 183

• the smallest number n such that φ(n) = 7! (see the number 779).

5 186

• the only number n 5 ·

109

such that φ(n) = φ(n + 1) = φ(n + 2): here the

common value is 2 592 =

25

·

34

(see R.K. Guy [101],

B36)149;

• the

16th

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

149Observe

that 5 186 = 2 · 2593, 5 187 = 3 · 7 · 13 · 19, 5 188 =

22

· 1297 and that φ(2593) = 2592 =

25 · 34 and φ(1297) = 1296 = 24 · 34, so that each odd prime number which shows up as a factor of

the numbers n, n + 1 and n + 2 is of the form

2α

·

3β

+ 1. One could ask the same question about

equations σ(n) = σ(n + 1) = σ(n + 2); note that if there is a solution, it is larger than 5 · 109.