Those Fascinating Numbers 61

n7 = 965 009 045; on the other hand, one can show71 that n8 5 708 691 485,

while n8 ≤ n = 65 893 166 030 = 2 · 5 · 17 · 19 · 23 · 43 · 47 · 277 since n + 1 =

3 · 7 · 11 · 29 · 31 · 41 · 71 · 109 (see the number 135 for the analogue problem with

the Ω(n) function).

231 (= 3 · 7 · 11)

• the smallest square-free composite number n such that p|n =⇒ p + 9|n + 9 (see

the number 399).

233

• the sixth prime Fibonacci number (see the number 89);

• the smallest prime factor of the Mersenne number 229 − 1, whose complete

factorization is given by

229

− 1 = 536 870 911 = 233 · 1103 · 2089;

the number

229

−1 is the smallest Mersenne number having three distinct prime

factors (see the number 223).

234

• the largest solution n 109 of

σ(n)

n

=

7

3

: the smallest one is n = 12.

235

• the fourth three digit number = 100, 200, 300, and whose square contains no

more than two distinct digits:

2352

= 55 225 (see the number 109).

236

• the number n which allows the sum

m≤n

ω(m)=2

1

m

to exceed 2 (see the number 44).

237

• the fourth number n such that

2n−1

+ n is a prime number: the only numbers

n 40 000 satisfying this property are 1, 3, 7, 237 and 1 885.

71It is clear that, for each number k 1, (nk + 1)2 nk(nk + 1) ≥ p1p2 . . . p2k, so that

nk ≥

√

p1p2 . . . p2k, from which it follows that n8 ≥ 5 708 691 485, and moreover that the inequality

is strict once we verify that the number 5 708 691 485 does not have the required property.