26 Jean-Marie De Koninck
81
the only number n 1 whose sum of digits is equal to

n;
the smallest number which can be written as the sum of three cubes and as the
sum of four cubes: 81 =
33
+
33
+
33
=
13
+
23
+
23
+
43.
82
the smallest number which can be written as the sum of four cubes as well as
the sum of five cubes: here 82 =
13
+
33
+
33
+
33
=
13
+
13
+
23
+
23
+
43.
83
the smallest number n which allows the sum
i≤n
1
i
to exceed 5: John V. Baxley
[19] developed a clever method to compute this number n; if nk stands for
the smallest number n which allows the sum
i≤n
1
i
to exceed k, we have the
following table:
k nk
2 4
3 11
4 31
5 83
6 227
7 616
8 1674
k nk
9 4550
10 12367
11 33617
12 91380
13 248397
14 675214
15 1835421
k nk
16 4989191
17 13562027
18 36865412
19 100210581
20 272400600
21 740461601
22 2012783315
the smallest number n which requires six iterations of the Euler φ function
to reach the number 2; in other words, n = 83 is the smallest number such
that φ6(n) = 2, where φk stands for the kth iteration of φ; the sequence of the
smallest numbers n = n(k), k = 1, 2, . . ., requiring k iterations of the φ function
in order to reach the number 2 begins as follows: 3, 5, 11, 17, 41, 83, 137, 257,
641, 1097, 2329, 4369, 10537, 17477, . . . (see R.K. Guy [101], B41);
the number appearing (as an exponent) in the diophantine equation
x83

y4871
= 1, an equation which cannot be solved by applying the “Inken criteria”,
the reason being that
834870
1 (mod
48712)
and
48712
1 (mod
832),
an
obstacle which was bypassed by M. Mignotte in 1992.
84
the smallest solution of
σ(n)
n
=
8
3
(see the number 1488);
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