Those Fascinating Numbers 133

1 618

• the integer part131 of

ee2

.

1 627

• the prime number which appears the most often as the

12th

prime factor of an

integer (see the number 199);

• the smallest number n for which π(n)

3

j=1

(j − 1)!n

logj

n

, this last expression

representing the first three terms of the asymptotic expansion of Li(n): here

π(1627) = 258 while

n

log n

+

n

log2 n

+

2n

log3 n

n=1627

≈ 257.832 (see the number

73);

• the ninth prime number p such that

(3p

− 1)/2 is itself a prime number (see

the number 1 091).

1 634

• the second number which can be written as the sum of the fourth powers of its

digits: 1 634 =

14

+

64

+

34

+

44;

the others are 1, 8 208 and 9 474;

• the sixth solution of σ(n) = σ(n + 1) (see the number 206).

1 638

• the fourth number which is neither perfect nor multi-perfect but whose har-

monic mean is an integer (see the number 140);

• the fourth solution of

σ(n)

n

=

8

3

(see the number 1 488).

1 639

• the 1

000th

square-free number (see the number 165).

131The

interest for this number stems from a number theory result according to which the

kth

prime factor qk (n) of a number n is “usually” of the order of

eek

, in the sense that, for all ε 0

and any function ξ(n) which tends to +∞ as n → ∞,

sup

ξ(n)≤k≤ω(n)

log log qk(n) − k

2k log k

≤ 1 + ε

almost everywhere. This result was first stated by P. Erd˝ os in 1946 (see G. Tenenbaum [193], p. 344).

Thus, q2 ≈ 1618, q3 ≈ 528 491 311 and q4 ≈ 514 843 556 263 450 564 886 528.