Those Fascinating Numbers 27

• the smallest number n 1 which divides σ8(n); the sequence of numbers satis-

fying this property begins as follows: 84, 156, 204, 364, 476, 514, 1092, 1428,

2316, 2652, 2892, 6069, 6188, 6748,. . . 32

85

• the smallest Smith number n (see the number 22) such that n − 1 is also a

Smith number.

86

• the largest known number k such that the decimal expansion of 2k does not

contain 0 as a digit; indeed, using a computer, one can verify that

286

= 77371252455336267181195264

does not contain the digit 0, while each number

2k,

for k = 87, 88, . . . , 3000

contains it (see the number 71).

87

• the smallest solution n 1 of σ(φ(n)) = σ(n); the

sequence33

of numbers

satisfying this equation begins as follows: 1, 87, 362, 1 257, 1 798, 5 002, 9 374,

21 982, 22 436, 25 978, . . .

88

• the total number of narcissistic numbers: an r digit number is said to be

narcissistic if it is the sum of the rth powers of its digits: hence if d1, d2, . . . , dr

are the r digits of such a number n, then n = d1 r + d2 r + . . . + dr; r here is the

list34 of all narcissistic numbers:

32It

is worth mentioning that it is known since Erd˝ os that for each integer k ≥ 2, there exist

infinitely many numbers n such that n|σk(n) (see Problem 11090, Amer. Math. Monthly 113

(2006), 372-373).

33It

would be interesting if one could prove that this equation does indeed have infinitely many

solutions. Observe on the other hand that the corresponding equation φ(σ(n)) = φ(n) could be

proved to have infinitely many solutions if one could prove that there exist infinitely many prime

numbers p = 2q − 1, where q is prime: indeed, if n is of the form n = 2p with p prime and if

p = 2q − 1 with q prime, q 3, then φ(σ(n)) = φ(3(p + 1)) = φ(3 · 2 · q) = 2(q − 1) while

φ(n) = φ(p) = p − 1 = 2(q − 1). De Koninck and Luca (see [57]) have shown that the number of

integers n ≤ x such that σ(φ(n)) = σ(n) is

x/(log2

x).

34In order to find the r digit narcissistic numbers (given that such numbers exist!) for a given

positive integer r ≤ 39 (and thus construct the table below), we proceeded as follows. Let n be the

quantity a11r +a22r +. . .+a99r , where the ai’s are non negative integers such that a1 +a2 +. . .+a9 =

r. If this number n has exactly r digits and if it is equal to the sum of the

rth

powers of its digits,

then it qualifies as a narcissistic number.