Those Fascinating Numbers 91
504
the smallest number m for which equation σ(x) = m has94 exactly ten solutions,
namely 204, 220, 224, 246, 284, 286, 334, 415, 451 and 503.
510
the smallest Niven number n 7 such that n + 1, n + 2 and n + 3 are also
Niven numbers (see the number 110).
512
(=83)
the smallest number n 1 whose sum of digits is equal to
3

n: the numbers
satisfying this property are 1, 512, 4 913, 5 832, 17 576 and 19 683;
the
13th
number n such that n ·
2n
1 is prime (see the number 115).
516
the fourth solution of σ(φ(n)) = φ(σ(n)); the sequence of numbers satisfying
this equation begins as follows: 9, 225, 242, 516, 729, 3 872, 13 932, 14 406,
17 672, 18 225, 20 124, 21 780, 29 262, 29 616, 45 996, 65 025, . . . 95
518
the fourth number n 9 such that n =
∑r
i=1
di,
i
where d1, . . . , dr stand for the
digits of n: here 518 =
51
+
12
+
83
(see the number 175).
521
the exponent of the
13th
Mersenne prime
2521
1 (Robinson, 1952).
528
the only number n for which the set A := {1, 8, 15, n} is a diophantine quadru-
plet, meaning that it is such that xy + 1 is a perfect square for all x, y A,
x = y (see the number 120 as well as its footnote).
94It
would be interesting if one could prove that the number of solutions x of σ(x) = m can be
arbitrarily large.
95It
is easy to show that if p and
(3p
1)/2 are two prime numbers, then n =
3p−1
is a solution
of σ(φ(n)) = φ(σ(n)) (S.W. Golomb [93]), in which case one obtains that n = 3p−1 is a solution for
p = 3, 7, 13, 71, 103, 541. On the other hand, to answer a question raised by Golomb in the same
paper, J.M. De Koninck & F. Luca [55] showed that, for each number u [0, 1], the density of the
set of numbers n such that σ(φ(n))/φ(σ(n))
ue2γ
(log log log
n)2
is strictly decreasing, varies in a
continuous manner (with respect to u) and is 0 when u = 1.
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