112 Jean-Marie De Koninck

916

• the only composite number n 108 such that σ(n + 22) − σ(n) = 22.

919

• the smallest number whose cube is the sum of two 3-powerful numbers: indeed,

9193

=

2713

+

23

·

35

·

733

(see the number 776 151 559).

923

• the seventh Hamilton number: the sequence (hn)n≥1 of Hamilton numbers can

be generated in a recursive manner by setting

h1 = 1, h2 = 2, hn = 2 +

n−1

j=2

(−1)j

j!

j−1

k=0

(hn+1−j − k) (n ≥ 3);

this sequence begins as follows: 1, 2, 3, 5, 11, 47, 923, 409 619, 83 763 206 255,. . .

930

• the smallest number n such that

φ(n)

n

=

8

31

; the sequence of numbers satis-

fying this equation begins as follows: 930, 1860, 2790, 3720, 4650, 5580, 7440,

8370, 9300, . . .

933

• (probably) the largest number which cannot be written as the sum of two

numbers whose index of composition is ≥ 3/2: in other words, if 933 = a + b,

then min(λ(a), λ(b)) 3/2; if nρ stands for the largest number n which cannot

be written as n = a + b with min(λ(a), λ(b)) ≥ ρ, then we have the following

table (all the values are based on numerical computations and are therefore

only conjectured values)118:

ρ 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

nρ 23 23 23 119 933 2 351 8 638 62 471 1 109 549 ? ?

935 (= 5 · 11 · 17)

• the second Lucas-Carmichael number (see the number 399);

• the smallest square-free composite number n such that p|n =⇒ p + 10|n + 10

(see the number 399).

118In a related matter, V. Blomer [22] established that the number of numbers ≤ x which can be

written as the sum of two powerful numbers (thus in particular with an index of composition ≥ 2)

is x/(log x)0.253.