222 Jean-Marie De Koninck

55 440

• the tenth colossally abundant number: we say that a number n is colossally

abundant if there exists ε 0 such that the function

σ(m)/m1+ε

reaches its

maximum at n (see P. Erd˝ os and J.L. Nicolas [78]); the sequence of numbers

satisfying this property begins as follows: 1, 2, 6, 12, 60, 120, 360, 2 520, 5 040,

55 440, 720 720, 1 441 440, 4 324 320, 21 621 600, 367 567 200, . . . ;

• the

28th

highly composite number (see the number 180).

56 000 (=

26

·

53

· 7)

• the sixth number n having at least two distinct prime factors and such that

B1(n) =

β(n)2:

here

26

+

53

+ 7 = (2 + 5 +

7)2

(see the number 144).

57 121 (= 2392)

• the sixth number n divisible by a square 1 and such that δ(n + 1) − δ(n) = 1

(see the number 49);

• the sixth solution w + s of the aligned houses problem (see the number 35).

57 122

• the seventh number n such that the binomial coeﬃcient

(n)

2

is a perfect square:

here

(

57 122

2

)

= 134 · 2392 (see the number 289).

57 860

• the fourth number n such that each of the numbers n + i, i = 0, 1, 2, . . . , 16, has

a factor in common with the product of the other 16 (see the number 2 184).

58 786

• the

11th

Catalan number (see the number 14).

58 980

• the number of twin prime pairs

107

(see the number 1 224).

59 400

• the second number n such that φ(n) + σ(n) = 4n (see also the numbers 23 760

and 312).