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.