52 Jean-Marie De Koninck
k nk nk
4 8 8
5 13 13
6 22 22
7 38 37
8 63 63
9 105 105
10 177 177
11 296 296
12 495 495
k nk nk
13 828 828
14 1386 1386
15 2318 2318
16 3879 3878
17 6489 6488
18 10854 10854
19 18158 18157
20 30375 30374
21 50811 50811
k nk nk
22 84998 84998
23 142187 142187
24 237853 237853
25 397885 397885
26 665589 665589
27 1113411 1113410
28 1862534 1862534
29 3115683 3115683
30 5211973 5211973
178
the eighth number n for which the distance from
en
to the nearest integer is the
smallest, that is for which κ(i) κ(n) for all i n, where κ(n) := min
m≥1
|en
m|:
the sequence of these numbers n begins as follows: 1, 3, 8, 19, 45, 75, 135, 178,
209, 732, 1351, 1907, . . .
180
the
11th
highly composite number: a number n is said to be highly composite
if τ (n) τ (m) for all numbers m n: here τ (180) = 18; the sequence of
numbers satisfying this property begins as follows: 1, 2, 4, 6, 12, 24, 36, 48, 60,
120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160,
25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200,
332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880,
3603600, 4324320, 6486480, 7207200, 8648640, . . . (see J.L. Nicolas [150]);
the only solution n
109
of σ(n) = 3n + 6;
the smallest number n 1 such that σ(n) and σ2(n) have the same prime
factors, that is such that γ(σ(n)) = γ(σ2(n)): here the common factors are 2,
3, 7 and 13; the sequence of numbers satisfying this property begins as follows:
180, 1444, 12996, 23805, 36100, 52020, 60228, 64980, 68832, 95220, 301140,
324900, 344160, 481824, . . .
181
the largest integer solution x of equation
x2
+ 7 =
2n:
in 1960, Nagell (see
L.J. Mordell [144]) established that this diophantine equation has only five so-
lutions, namely (x, n) = (1, 3), (3,4), (5,5), (11,7) and (181,15).
182
the smallest solution of σ(n) = σ(n + 13); the sequence of numbers satisfying
this equation begins as follows: 182, 782, 1965, 2486, 2678, 2685, 12141, 12441,
17342, 21242, . . .
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