36 Jean-Marie De Koninck

110

• the solution c of the diophantine equation

a5 +b5 +c5 +d5

=

e5

which, according

to Euler, had no solutions: the solution

(a, b, c, d, e) = (27, 84, 110, 135, 144)

was discovered by L. Lander & T. Parkin in 1966;

• the smallest Niven number n 9 such that n + 1 and n + 2 are also Niven

numbers: we say that n is a Niven number if it is divisible by the sum of its

digits; if we

let43

nk stand for the smallest Niven number n 9 such that

n + 1, n + 2, . . . , n + k − 1 are also Niven numbers, then n2 = 20, n3 = 110,

n4 = 510, n5 = 131 052, n6 = n7 = 10 000 095 and n8 = n9 = 124 324 220.

111

• the smallest insolite number: a number 1 is called insolite if it does not

contain the digit 0 in its decimal expansion and if it is divisible by both the sum

and the product of the squares of its digits: the sequence of insolite numbers is

infinite44

and begins as follows: 111, 11 112, 1 122 112, 111 111 111, 122 121 216,

1 111 112 112, 1 111 211 136, 1 116 122 112, 1 211 162 112, 11 111 113 116,

11 111 121 216, 11 112 122 112, 11 121 114 112, 11 132 111 232, 11 133 122 112,

11 213 111 232, 11 311 322 112, 12 111 213 312, 21 111 311 232, 31 111 221 312,

32 111 111 232, 111 122 111 232, 111 132 122 112, 111 211 322 112,

111 312 122 112, 112 111 322 112, 113 112 122 112, 121 111 216 128,

121 111 322 112, 121 121 114 112, 131 111 132 112, 131 112 122 112,

211 111 322 112, 211 121 114 112, 311 112 122 112, 911 131 213 824,

1 111 111 113 312, 1 111 121 114 112, 1 121 313 321 216, 1 331 611 322 112, . . . ; the

following table provides, for each number k ≤ 9, the list of the smallest insolite

number n = nk containing the digit k:

43It

has been known at least since 1997 (see Wilson [207]) that it is possible to construct a sequence

of 20 consecutive Niven numbers, but that no strings of 21 consecutive Niven numbers exist. More

recently, in 2008, De Koninck, Doyon & K´ atai [51] obtained, for each positive integer r ≤ 20, an

asymptotic formula for the number of r-tuples (n, n + 1, . . . , n + r − 1), where each n + i is a Niven

number, with n ≤ x.

44It

is easy to see that any number 11 . . . 1,

k

where k is such that

k|(10k

− 1), is such a number,

which is the case when k = 3, 9, 27, 81, 111, 243, 333, 729, 999, . . . But this last sequence is infinite

because it contains all numbers of the form

3α,

the reason being that

3α

divides

103α

− 1 for each

number α ≥ 1, a result which can easily be proved by induction. J.M. De Koninck & N. Doyon [48]

proved that if I(x) stands for the number of insolite numbers ≤ x, then

exp

1

5

(log log

x)2

+ O(log log x log log log x) I(x)

x0.462.