298 Jean-Marie De Koninck

12 792 675

• the sixth number n such that β(n)|β(n + 1) and β(n + 1)|β(n + 2): here 74|222

and 222|412698 (see the number 225 504).

12 870 973

• the smallest number n such that π(n)

5

j=1

(j − 1)!n

logj

n

, this last expression

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

π(12870973) = 841464 while

∑5

j=1

(j−1)!n

logj n

n=12870973

≈ 841463.3 (see the

number 73).

12 999 168 (=

29

·

32

· 7 · 13 · 31)

• the fourth solution of

σ(n)

n

=

11

3

(see the number 35 640).

13 053 769

• the smallest perfect square m5 2 for which there exist numbers m1, m2, m3 and

m4 such that mi 2 −(mi−1)2 = mi−1 2 for i = 2, 3, 4, 5: here 13 053 769 = 36132 =

36122 + 852 = 36122 + 842 + 132 = 36122 + 842 + 122 + 52 = 36122 + 842 +

122 + 42 + 32.

13 141 793

• the smallest number n such that σ(n), σ(n + 1), . . . , σ(n + 5) all have the same

prime factors, namely here the primes 2, 3, 5, 7 and 149 (see the number 3 777).

13 466 917

• the exponent of the

39th

Mersenne prime

213 466 917

− 1 discovered by the Cana-

dian Michael Cameron in November 2001 using the programme developed by

G. Woltman (see the number 1 398 269).

13 562 027

• the smallest number n which allows the sum

i≤n

1

i

to exceed 17 (see the number

83).