Those Fascinating Numbers 153
3 059
the smallest number n such that π(n) = n/7 (see the number 330).
3 114
the only four digit number (= 1000, 2000, 3000) whose square contains only two
distinct digits: 3 1142 = 9 696 996 (see the number 109).
3 126 (= 2 · 3 · 521)
the fifth
number143
of the form
nn
+ 1.
3 136 (= 2 · 6 ·
72)
the 100th powerful number; if nj stands for the jth powerful number, then we
have the following table (here 1 counts as a powerful number):
k n10k
1 49
2 3 136
3 253 472
k n10k
4 23 002 083
5 2 200 079 025
6 215 523 459 072
k n10k
7 21 348 015 504 200
8 2 125 390 162 618 116
9 212 104 218 976 916 644
3 159 (= 35 · 13)
the third number n having at least two distinct prime factors and such that
B1(n) = β(n)2: here 35 + 13 = (3 + 13)2 (see the number 144).
3 160
the
11th
number n such that f(n) f(m) for all numbers m n, where
f(n) :=
Σ∗
1
p
, where the star indicates that the sum runs through all prime
numbers p n which do not divide
(
2n
n
)
: here f(3160) = 1.11552 . . . (see the
number 364).
143One
can prove that the number
nn
+ 1 is composite for 5 n
264.
Indeed, if n is odd, then
nn
+1 is even. On the other hand, if n is even but not a power of 2, let s be the largest positive
such that
2s
n, so that n =
2s
·m where m 1 is odd; in this case,
nn
+1 =
n2s·m
+1 =
(
n2s
)minteger
+1,
in which case
n2s
+1 is a proper divisor of
nn
+1, thus implying that
nn
+1 is composite. It remains
to consider the case where n is a power of 2. In this case, n =
2s
and
nn
+1 =
(2s)2s
+1 =
2s·2s
+1.
If s is not a power of 2, it is easy to see by using once more the above argument that
nn
+ 1 is
composite. On the other hand, if s is a power of 2, there exists a positive integer r such that
nn
+ 1 =
22r
+ 1 = Fr , that is a Fermat number, and this happens when n =
22β
and r = β +

,
in which case we find
44
+ 1 = F3,
1616
+ 1 = F6,
256256
+ 1 = F11,
(216)216
+ 1 = F20,
(232)232
+ 1 = F37.
But it is known that the Fermat numbers F6, F11, F20 and F37 are all composite. Hence, it follows
that nn + 1 is composite for each 5 n 264.
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