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).
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