Those Fascinating Numbers 97

619

• the number of powerful numbers ≤

105

(we assume here that 1 is a powerful

number); if ρ(x) stands for the number of

powerful100

numbers ≤ x, then we

have the following table:

x ρ(x)

10 4

102 14

103 54

104 185

105

619

106

2 027

107

6 553

x ρ(x)

108 21 044

109 67 231

1010 214 122

1011 680 330

1012

2 158 391

1013

6 840 384

1014

21 663 508

x ρ(x)

1015 68 575 557

1016 217 004 842

1017 686 552 743

1018 2 171 766 332

1019

6 869 227 848

1020

21 725 636 644

1021

68 709 456 167

623

• possibly the largest number n such that n(n + 1)(n + 2) has exactly the same

prime factors as m(m + 1)(m + 2) for a certain number m n: here m = 89

and the prime factors common to these two quantities are 2, 3, 5, 7, 13 and 89,

since

89 · 90 · 91 = 2 ·

32

· 5 · 7 · 13 · 89

623 · 624 · 625 =

24

· 3 ·

53

· 7 · 13 · 89

(see the number 340).

625

• the smallest fourth power which can be written as the sum of five fourth powers:

625 =

54

=

24

+

24

+

34

+

44

+

44;

• the smallest three digit automorphic number: 6252 = 390 625 (see the number

76).

627

• the number of digits in the fifth prime number whose digits are 1 and 2 in

alternation, that is a prime number of the form 1212 . . . 121 (see the number

139).

100In

order to compute ρ(x), we use the fact that each powerful number can be written in a unique

manner as m3r2, where m is square-free, so that we have

ρ(x) =

m3r2≤x

µ2(m)

=

m≤x1/3

µ2(m)

x

m3

.