82 Jean-Marie De Koninck
k ρ(k)
2 2
4 13
5 5
7 7
8 17
10 19
11 29
13 67
14 59
16 79
17 89
k ρ(k)
19 199
20 389
22 499
23 599
25 997
26 1889
28 1999
29 2999
31 4999
32 6899
34 17989
k ρ(k)
35 8999
37 29989
38 39989
40 49999
41 59999
43 79999
44 98999
46 199999
47 389999
49 598999
50 599999
k ρ(k)
52 799999
53 989999
55 2998999
56 2999999
58 4999999
59 6999899
61 8989999
62 9899999
64 19999999
65 29999999
67 59899999
k ρ(k)
68 59999999
70 189997999
71 89999999
73 289999999
74 389999999
76 689899999
77 699899999
79 799999999
80 998999999
82 2999899999
83 3999998999
390 (= 2 · 3 · 5 · 13)
the fifth ideal number: an r digit number is said to be an ideal number if its
jth factor, for each 1 j r, is the one that appears the most often as the
jth prime factor of an integer (see the number 199): the first 20 ideal numbers
are: 2, 6, 30, 42, 390, 546, 8970, 12558, 421590, 590226, 47639670, 66695538,
9480294330, 13272412062, 682923295390, 3756092613546, 1252925178947130,
1754095250525982, 1111344633726104310 and 1555882487216546034;
the smallest number which cannot be written as the sum of seven non zero
distinct squares (R.K. Guy [101], C20).
392
the only solution n
1012
of σ(n) = 2n + 71 (see the number 196).
396
the third number n = [d1, d2, . . . , dr] such that (d1 +1)·(d2 +2)·. . .·(dr +r) = n;
the sequence of numbers satisfying this property is infinite and includes the
numbers 16, 27, 396, 117 729 612 000, 266 744 016 000, 722 905 497 600,
1 234 057 144 320 and 16 231 012 761 600.
399 (= 3 · 7 · 19)
the
14th
number n such that n! + 1 is prime (see the number 116);
the smallest solution of τ (n + 1) τ (n) = 7; the sequence of numbers satisfying
this equation begins as follows: 399, 783, 2703, 3249, 4623, 11024, 15129, 16383,
17689, . . . ; if nk stands for the smallest number n such that τ (n+1)−τ (n) = k,
then the sequence (nk)k≥1 begins as follows: 1, 5, 49, 11, 35, 23, 399, 47, 1849,
59, 143, 119, 1599, 167, 575, 179, 1295, 239, 4355, 629, 2303, 359, 899, 959,
9215, 1007, 39999, 719, 20735, 839, . . . ;
the largest number of the form 8k + 7 which can be written as the sum of
exactly three powerful numbers in only one way: here 399 = 49 + 125 + 225
(see the number 118);
Previous Page Next Page