Those Fascinating Numbers 149
2 600
the number of numbers 10 000 which are the product of two distinct prime
numbers (and thus square-free); if we denote by πk(x) the number of numbers
x which are the product of k distinct prime numbers (and thus square-free),
we have the following table:141
x π(x) π2(x) π3(x) π4(x)
10 4 2 0 0
102
25 30 5 0
103
168 288 135 16
104 1229 2600 1800 429
105 9592 23313 19919 7039
106 78498 209867 206964 92966
107 664579 1903878 2086746 1103888
108 5761455 17426029 20710806 12364826
109 50847534 160785135 203834084 133702610
1010 455052511 1493766851 1997171674 1413227318
1011 4118054813 13959793240 19675145130 15051992868
1012 37607912018 130330450475 191186911415 155488013308
2 609
the eighth number n such that n! + 2n 1 is prime (see the number 6 247).
2 625
the 12th solution of φ(n) = φ(n + 1) (see the number 15).
2 636
the sixth number which is not a palindrome, but whose square is a palindrome
(see the number 26).
2 656
the largest number n such that π(n) Li(n)

n log n

provided the Riemann
Hypothesis is true (see L. Schoenfeld [182]).
141To
construct this table, one can proceed as follows. Let x be a sufficiently large number. Given
two numbers k 2 and r
x1/k,
denote by πk(x, r) the number of numbers x which are the
product of k distinct prime numbers, the smallest being larger than r. Then, use the fact that
π1(x) = π(x) and that, for each k 2,
πk(x) =
px1/k
πk−1
x
p
, p .
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