Those Fascinating Numbers 219

seems that there does not exist any number (= prime power) equal to the sum

of the squares of its prime factors163.

46 233

• the value of 1! + 2! + . . . + 8!.

46 657 (= 13 · 37 · 97)

• the sixth number of the form

nn

+ 1 (see the number 3 126);

• the second pseudoprime in bases 2, 3, 5 and 7 (see the number 29 341).

47 196

• the fifth number n such that σ3(n) is a perfect square: indeed we have σ3(47 196) =

111602402 (see the number 345).

47 293

• the seventh horse number (see the number 13).

47 616 (=

29

· 3 · 31)

• the smallest solution of

σ(n)

n

=

11

4

; the second solution of this equation is

134 209 536.

48 625

• the smallest number n = [d1, d2, . . . , dr] such that n = d1r

d

+ d2r−1

d

+ . . . + dr1

d

:

here 48 625 =

45

+

82

+

66

+

28

+

54;

the only other known number satisfying

this property is 397 612: this is an observation due to Patrick De Geest.

48 947

• the fourth prime number p such that 17p−1 ≡ 1 (mod p2) (see the number

46 021).

49 081

• the sixth number k such that 11 . . . 1

k

is prime (Granlund & Dubner, 1989); see

the number 19.

163In

fact, one can prove that if such a number exists, then it must have at least five distinct prime

factors, and moreover that if it is square-free, then it must have at least 13 prime factors.