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