Those Fascinating Numbers 165
5 044 (= 22 · 13 · 97)
the smallest number n such that n, n + 2, n + 4, . . . , n + 10 are all divisible by
a square 1.
5 101
the fifth odd number k such that 2n + k is composite for all n k (see the
number 773): in fact, 2n + 5 101 is composite for each n 5 759 and prime for
n = 5 760.
5 104
the smallest number which can be written as the sum of three distinct cubes
in three distinct ways:
5 104 =
13
+
123
+
153
=
23
+
103
+
163
=
93
+
103
+
153
(see the number 1 009).
5 105
the largest number k such that inequality θ(pk) k(log k +log log k −1) (where
θ(x) =

p≤x
log p) is false (G. Robin [175]).
5 120 (=
210
· 5)
the smallest number n having at least two distinct prime factors and such that
β(n)3|B1(n):
here (2 +
5)3|(210
+ 5); the only numbers
108
satisfying this
property are 5 120, 419 904, 885 735, 5 315 625 and 18 003 384.
5 183
the smallest number n such that φ(n) = 7! (see the number 779).
5 186
the only number n 5 ·
109
such that φ(n) = φ(n + 1) = φ(n + 2): here the
common value is 2 592 =
25
·
34
(see R.K. Guy [101],
B36)149;
the
16th
solution of φ(n) = φ(n + 1) (see the number 15).
149Observe
that 5 186 = 2 · 2593, 5 187 = 3 · 7 · 13 · 19, 5 188 =
22
· 1297 and that φ(2593) = 2592 =
25 · 34 and φ(1297) = 1296 = 24 · 34, so that each odd prime number which shows up as a factor of
the numbers n, n + 1 and n + 2 is of the form

·

+ 1. One could ask the same question about
equations σ(n) = σ(n + 1) = σ(n + 2); note that if there is a solution, it is larger than 5 · 109.
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