Those Fascinating Numbers 153

3 059

• the smallest number n such that π(n) = n/7 (see the number 330).

3 114

• the only four digit number (= 1000, 2000, 3000) whose square contains only two

distinct digits: 3 1142 = 9 696 996 (see the number 109).

3 126 (= 2 · 3 · 521)

• the fifth

number143

of the form

nn

+ 1.

3 136 (= 2 · 6 ·

72)

• the 100th powerful number; if nj stands for the jth powerful number, then we

have the following table (here 1 counts as a powerful number):

k n10k

1 49

2 3 136

3 253 472

k n10k

4 23 002 083

5 2 200 079 025

6 215 523 459 072

k n10k

7 21 348 015 504 200

8 2 125 390 162 618 116

9 212 104 218 976 916 644

3 159 (= 35 · 13)

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

B1(n) = β(n)2: here 35 + 13 = (3 + 13)2 (see the number 144).

3 160

• the

11th

number n such that f(n) f(m) for all numbers m n, where

f(n) :=

Σ∗

1

p

, where the star indicates that the sum runs through all prime

numbers p n which do not divide

(

2n

n

)

: here f(3160) = 1.11552 . . . (see the

number 364).

143One

can prove that the number

nn

+ 1 is composite for 5 ≤ n

264.

Indeed, if n is odd, then

nn

+1 is even. On the other hand, if n is even but not a power of 2, let s be the largest positive

such that

2s

n, so that n =

2s

·m where m 1 is odd; in this case,

nn

+1 =

n2s·m

+1 =

(

n2s

)minteger

+1,

in which case

n2s

+1 is a proper divisor of

nn

+1, thus implying that

nn

+1 is composite. It remains

to consider the case where n is a power of 2. In this case, n =

2s

and

nn

+1 =

(2s)2s

+1 =

2s·2s

+1.

If s is not a power of 2, it is easy to see by using once more the above argument that

nn

+ 1 is

composite. On the other hand, if s is a power of 2, there exists a positive integer r such that

nn

+ 1 =

22r

+ 1 = Fr , that is a Fermat number, and this happens when n =

22β

and r = β +

2β

,

in which case we find

44

+ 1 = F3,

1616

+ 1 = F6,

256256

+ 1 = F11,

(216)216

+ 1 = F20,

(232)232

+ 1 = F37.

But it is known that the Fermat numbers F6, F11, F20 and F37 are all composite. Hence, it follows

that nn + 1 is composite for each 5 ≤ n 264.