Those Fascinating Numbers 147
2 427
the eighth number n 9 such that n =
∑r
i=1
di,
i
where d1, . . . , dr stand for the
digits of n: here 2 427 =
21
+
42
+
23
+
74
(see the number 175).
2 430
the third number n 2 such that
σ(n) + φ(n)
γ(n)2
is an integer (see the number
588).
2 465 (= 5 · 17 · 29)
the fourth Carmichael number (see the number 561);
the value of the sum of the elements of a diagonal, of a line or of a column in
a 17 × 17 magic square (see the number 15).
2 467
the number of digits in the decimal expansion of the Fermat number
2213
+ 1.
2 474
the second number n such that

p≤pn
p is a perfect square; the only140 known
numbers n for which

p≤pn
p is a perfect square are 9, 2 474, 6 694, 7 785,
709 838 and 126 789 311 423.
2 491
the third odd number k such that
2n
+ k is composite for all numbers n k
(see the number 773): in fact,
2n
+ 2 491 is composite for each n 3 536 and
prime for n = 3 536.
2 499
the fifth number n such that
2n
7 is prime (see the number 39).
2 500
the fourth perfect square which is a Smith number (see the number 22): 2 500 =
22
·
54
and 2 + 5 = 7 = 2 + 5; the three smallest are 361, 1 600 and 2 401.
140F.
Luca [128] proved that the set T of numbers n such that

p≤pn
p is a perfect square is of
zero density. He also provided a heuristic argument which suggests that T is an infinite set while
each of the sets Tk := {n :

p≤pn
p is a kth power}, for k 3, is a finite set. Luca’s argument
suggests that #{n x : n T } 2 log x, while numerical evidence seem to indicate that each
set Tk, for k 3, is empty.
Previous Page Next Page