206 Jean-Marie De Koninck

26 208 (= 25 · 32 · 7 · 13)

• the third solution of

σ(n)

n

=

7

2

(see the number 4 320).

26 277 (= 3 · 19 · 461)

• the number n which allows the sum

m≤n

ω(m)=3

1

m

to exceed 2 (see the number

1 953).

26 608 (=

24

· 1663)

• the 10

000th

number having exactly two distinct prime factors (see the number

184).

26 861

• the smallest number n for which π(n; 4, 1) π(n; 4, 3) (Leech, 1957).

26 951

• the

19th

number n (and the largest one known) such that n! + 1 is prime (see

the number 116); note that the number 26 951! + 1 is a 107 707 digit number.

27 000

• the fifth number n 2 such that

σ(n) + φ(n)

γ(n)2

is an integer (see the number

588).

27 625 (= 53 · 13 · 17)

• the smallest number which can be written as the sum of two squares in seven

distinct ways (as well as in eight distinct ways), namely 27 625 =

202

+

1652

=

272 +1642 = 452 +1602 = 602 +1552 = 832 +1442 = 882 +1412 = 1012 +1322 =

1152 + 1202 (see the number 50).

27 720 (= 23 · 32 · 5 · 7 · 11)

• the smallest number n such that σ(n) ≥ 4n: in this case,

σ(n)

n

≈ 4.05; the

sequence of numbers satisfying this property begins as follows: 27 720, 30 240,

32 760, 50 400, 55 440, 60 480, 65 520, 75 600, 83 160, . . . ; moreover, if n = nk

stands for the smallest number n such that σ(n) ≥ kn, then n2 = 6, n3 = 120,

n4 = 27 720, n5 = 122 522 400 and n6 = 130 429 015 516 800;

• the 25th highly composite number (see the number 180).