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.
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