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
.
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