Those Fascinating Numbers 155

3 391

• the smallest prime factor of the Mersenne number

2113

− 1, whose complete

factorization is given by

2113

− 1 = 3391 · 23279 · 65993 · 1868569 · 1066818132868207,

this number being the smallest Mersenne number having exactly five prime

factors (see the number 223);

• the smallest prime number having two representations of the form

x3

+

2y3:

here 3 391 =

153

+ 2 ·

23

=

93

+ 2 ·

113

(see the number 41 for important

references); the sequence of numbers satisfying this property begins as follows:

3 391, 8 317, 79 939, 593 209, 942 199, 1 229 257,. . .

3 413

• the second prime number of the form

∑n

i=1

ii

(here with n = 5); setting sj =

∑j

i=1

ii

and letting nk stand for the

kth

prime number in the sequence (sj )j≥1,

then n1 = s2 = 11 + 22 = 5, n2 = s5 = 3 413, n3 = s6 = 50 069, n4 = s10 =

10 405 071 317 and n5 = s30; no other prime numbers of this form are known144.

3 435

• the smallest number n = [d1, d2, . . . , dr] 1 such that

1≤i≤r

di=0

di

di

= n; the only

other number satisfying this property is n = 438 579 088.

3 445 (= 5 · 13 · 53)

• the smallest square-free composite number n such that p|n =⇒ p + 11|n + 11.

3 462

• the number of solutions 2 ≤ x1 ≤ x2 ≤ . . . ≤ x6 of

6

i=1

1

xi

= 1 (see the number

147, as well as R.K. Guy [101], D11).

144It

is easy to establish that all terms s4k−1 and s4k are even, thus not prime. In 1993,

K. Soundararajan [189] proved that if A = {s1, s2, . . .} and if πA(x) = #{p ≤ x : p ∈ A}, that

is the cardinality of the set of prime numbers p ≤ x which belong to A, then πA(x)

log x

(log log x)2

.

This same author conjectures that there exists a constant c 0 such that πA(x) ∼ c

log x

(log log x)2

,

and therefore in particular that there exist infinitely many prime numbers of the form

11

+

22

+

33 + . . . + nn.