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

8π

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 .