Those Fascinating Numbers 345

3 569 485 920 (= 25 · 3 · 5 · 7 · 11 · 13 · 17 · 19 · 23)

• the number n for which the expression

Ω(n)ω(n)

n

reaches its maximal value203,

namely 2.97088 (see the numbers 60 and 120 120).

3 663 002 302

• the number of 11 digit prime numbers (see the number 21).

3 736 935 913

• the smallest number n such that π(n)

6

j=1

(j − 1)!n

logj

n

, this last expression

representing the first six terms of the asymptotic expansion of Li(n): here

π(3 736 935 913) = 178 046 624, while

∑6

j=1

(j−1)!n

logj

n

n=3 736 935 913

≈ 178 046 623.82 (see the number 73).

3 778 888 999

• the smallest number of persistence 10 (see the number 679).

3 786 686 400

• the

14th

number n such that φ(n) + σ(n) = 4n (see the number 23 760).

4 044 619 541

• the smallest number n such that P (n) P (n + 1) . . . P (n + 12): here

4044619541 674103257 19168813 10317907 5737049 1447609

465809 451207 199429 162109 115399 82021 1913 (see the

number 1 851).

4 118 054 813

• the number of prime numbers 1011.

203It

is easy to establish that this number n is of the form n =

2a−1p1p2

. . . pk for certain positive

integers a and k; one can then prove that the optimal choice is obtained by setting a = 5 and k = 9.