234 Jean-Marie De Koninck
97 247
the largest number equal to the sum of the fifth powers of its digits added to
the product of its digits (see the number 1 324).
98 304
the 16th Granville number (see the number 126).
98 726
the 24th number n such that n · 2n 1 is prime (see the number 115).
98 731
the largest prime number made up of exactly five distinct digits (see the number
9 871).
99 066
the largest number whose square uses each of the ten digits once and only once:
87 numbers satisfy this property, the smallest being 32 043 (for the complete
list, see the number 32 043).
99 528
the smallest number n which is not a prime power, but which is such that
βi(n)|n for i = 1, 2, where βi(n) =

p|n
pi; if, for each positive integer k, nk
stands for the smallest number n which is not a prime power, but which is such
that βi(n)|n for i = 1, . . . , k, we then have the following table170:
k n = nk βi(n)
1 30 = 2 · 3 · 5 β1(n) = 10 = 2 · 5
2 99 528 =
23
· 3 · 11 · 13 · 29 β1(n) = 58 = 2 · 29
β2(n) = 1 144 =
23
· 11 · 13
3 12 192 180 =
22
· 3 · 5 ·
72
· 11 · 13 · 29 β1(n) = 70 = 2 · 5 · 7
β2(n) = 1 218 = 2 · 3 · 7 · 29
β3(n) = 28 420 =
22
· 5 ·
72
· 29
4 n4 n = 2078479331940068525081053440 β1(n) = 2 · 33 · 11
= 28 · 33 · 5 · 7 · 11 · 13 · 17 · 23 · 31 · 37 β2(n) = 22 · 7 · 17 · 71
·412
· 47 · 53 · 61 · 71 · 83 · 89 β3(n) =
23
· 3 · 31 · 37 · 83
(a 28 digit number) β4(n) =
28
· 17 · 23 ·
412
5 n5 n =
24
· 3 · 5 ·
74
· 11 · 13 · 23 · 29 · 31 β1(n) = 2 · 13 · 137
·41 · 43 · 47 · 53 · 67 · 73 · 79 · 83 · 89 · 97 β2(n) =
23
· 3 · 7 · 41 · 73
·101 · 103 · 107 · 109 · 113 · 127 · 131 β3(n) =
24
·
74
· 13 · 163
·137 · 151 · 163 · 167 · 173 · 179 β4(n) =
22
· 3 · 7 · 47 · 109 · 173 · 191
·181 · 191 · 199 · 211 · 223 β5(n) = 23 · 11 · 89 · 97 · 127 · 151 · 179
(a 70 digit number)
(see also the table given at the number 378).
170See J.M. De Koninck & F. Luca [56] for a proof that the number nk exists for each k.
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