196 Jean-Marie De Koninck
Base b minimal number n = [d1, d2, . . . , dr]b = d1! · d2! ··· dr!
5 144 = [1, 0, 3, 4]5 = 1! · 0! · 3! · 4! =
24
·
32
6 24 = [4, 0]6 = 4! · 0! =
23
· 3
7 248 832 000 = [6, 1, 1, 1, 0, 1, 5, 5, 0, 4]7
= 6! · 1! · 1! · 1! · 0! · 1! · 5! · 5! · 0! · 4! = 213 · 35 · 53
8 17 280 = [4, 1, 6, 0, 0]8 = 4! · 1! · 6! · 0! · 0! = 27 · 33 · 5
9 2 264 832 171 464 196 096 000 000
= [3, 1, 3, 4, 7, 6, 3, 6, 2, 1, 0, 4, 2, 1, 5, 5, 7, 4, 0, 0, 0, 0, 0, 0, 0, 0]9
= 236 · 316 · 56 · 72
10 (no numbers satisfying this property are known)
11 36 = [3, 3]11 = 3! · 3! = 22 · 32
12 21 772 800 = [7, 3, 6, 0, 0, 0, 0]12
= 7! · 3! · 6! · 0! · 0! · 0! · 0! =
29
·
35
·
52
· 7
17 316
the sixth number whose square can be written as the sum of three fourth powers:
17
3162
=
724
+
904
+
1204
(see the number 481).
17 576
(=263)
the fourth number n whose sum of digits is equal to
3

n (see the number 512).
17 850
the smallest solution of σ(n) = 3n + 18: the only solutions n 109 of this
equation are 17 850, 64 890 and 884 730.
18 095
the tenth Lucas-Carmichael number (see the number 399).
18 121
the smallest prime factor of the Mersenne number
2151
1, whose complete
factorization is given by
2151
1 = 18121 · 55871 · 165799 · 2332951 · 7289088383388253664437433.
18 158
the smallest number n which allows the sum
m≤n
1
φ(m)
to exceed 19 (see the
number 177).
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