Those Fascinating Numbers 147

2 427

• the eighth number n 9 such that n =

∑r

i=1

di,

i

where d1, . . . , dr stand for the

digits of n: here 2 427 =

21

+

42

+

23

+

74

(see the number 175).

2 430

• the third number n 2 such that

σ(n) + φ(n)

γ(n)2

is an integer (see the number

588).

2 465 (= 5 · 17 · 29)

• the fourth Carmichael number (see the number 561);

• the value of the sum of the elements of a diagonal, of a line or of a column in

a 17 × 17 magic square (see the number 15).

2 467

• the number of digits in the decimal expansion of the Fermat number

2213

+ 1.

2 474

• the second number n such that

∑

p≤pn

p is a perfect square; the only140 known

numbers n for which

∑

p≤pn

p is a perfect square are 9, 2 474, 6 694, 7 785,

709 838 and 126 789 311 423.

2 491

• the third odd number k such that

2n

+ k is composite for all numbers n k

(see the number 773): in fact,

2n

+ 2 491 is composite for each n 3 536 and

prime for n = 3 536.

2 499

• the fifth number n such that

2n

− 7 is prime (see the number 39).

2 500

• the fourth perfect square which is a Smith number (see the number 22): 2 500 =

22

·

54

and 2 + 5 = 7 = 2 + 5; the three smallest are 361, 1 600 and 2 401.

140F.

Luca [128] proved that the set T of numbers n such that

∑

p≤pn

p is a perfect square is of

zero density. He also provided a heuristic argument which suggests that T is an infinite set while

each of the sets Tk := {n :

∑

p≤pn

p is a kth power}, for k ≥ 3, is a finite set. Luca’s argument

suggests that #{n ≤ x : n ∈ T } ∼ 2 log x, while numerical evidence seem to indicate that each

set Tk, for k ≥ 3, is empty.