348 Jean-Marie De Koninck

6 210 001 000

• the only self descriptive number, that is whose

kth

digit starting with the left

(for k = 1, 2, . . . , 10) indicates the number of times that the digit k − 1 appears

in its decimal expansion (see Pickover [160], pp. 217-219).

6 329 788 416

• the smallest number not containing the digit 0, divisible both by the product

of its digits and by the sum of its digits, and containing the maximum possible

number of distinct digits, namely the digits 1,2,3,4,6,7,8,9 (compare with the

number 711 813 411 914 121 216).

6 564 120 420

• the

20th

Catalan number (see the number 14).

6 661 661 161

• the largest known perfect square containing only two distinct digits excluding

0; the only known perfect squares satisfying this property are 16, 25, 36, 49,

64, 81, 121, 144, 225, 441, 484, 676, 1444, 7744, 11881, 29929, 44944, 55225,

69696, 9696996 and 6661661161.

6 983 776 800 (= 25 · 33 · 52 · 7 · 11 · 13 · 17 · 19)

• the number n at which the expression

log τ (n) log log n

log 2 log n

reaches its maximal

value, namely 1.537939861 . . . (J.L. Nicolas & G. Robin [152]).

7 362 724 275

• the smallest integer n such that ω(n), ω(n + 1), . . . , ω(n + 7) are all distinct,

namely in this case with the values 4, 7, 1, 8, 2, 5, 6 and 3 (see the number

417).

7 725 038 629

• the smallest number x which contradicts the von Sterneck Conjecture accord-

ing to which |M(x)|

1

2

√

x (where M(x) :=

∑

n≤x

µ(n)) for x 200 (see

Ribenboim [169], p. 231): here M(7 725 038 629) = 43 947

1

2

√

7 725 038 629 ≈

43946.1 .