218 Jean-Marie De Koninck
45 125
the tenth powerful number which can be written as the sum of two co-prime
3-powerful numbers = 1: 45 125 = 20 736+24 389, that is 53 · 192 = 28 · 34 +293
(see the number 841).
45 360
the
26th
highly composite number (see the number 180).
45 864
(=23
·
32
·
72
· 13)
the fifth Erd˝ os-Nicolas number (see the number 2 016).
45 918
the smallest number n such that

m≤n
τ (m) is a multiple of 100 000 (here the
sum is equal to 500 000).
46 021
the third prime number p such that
17p−1
1 (mod
p2):
the only prime
numbers p
232
satisfying this congruence are 2, 3, 46 021 and 48 947 (see
Ribenboim [169], p. 347).
46 189
the smallest number n such that P (n) P (n + 1) . . . P (n + 7): here
19 149 173 2887 6559 7699 9239 11549; if we denote by nk the
smallest number n such that P (n) P (n + 1) . . . P (n + k 1), then we
have the following table:
k 3 4 5 6 7 8 9
nk 8 8 90 168 9 352 46 189 721 970
k 10 11 12
nk 721 970 6 449 639 565 062 156
(compare with the table given at number 714).
46 206 (= 2 ·
32
· 17 · 151)
the smallest number which is not a prime power, but which is divisible by the
sum of the squares of its prime factors: here 22 +32 +172 +1512 = 23103|46206;
the sequence of numbers satisfying this property begins as follows: 46 206,
72 105, 73 346, 92 412, 96 096, 97 440, 98 098, 99 528, 113 883, 117 040, 127 680,
134 805, 138 618, 143 520, 146 692, 150 024, 165 880, 165 886, 184 824, . . . : it
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