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 .
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