112 Jean-Marie De Koninck
916
the only composite number n 108 such that σ(n + 22) σ(n) = 22.
919
the smallest number whose cube is the sum of two 3-powerful numbers: indeed,
9193
=
2713
+
23
·
35
·
733
(see the number 776 151 559).
923
the seventh Hamilton number: the sequence (hn)n≥1 of Hamilton numbers can
be generated in a recursive manner by setting
h1 = 1, h2 = 2, hn = 2 +
n−1
j=2
(−1)j
j!
j−1
k=0
(hn+1−j k) (n 3);
this sequence begins as follows: 1, 2, 3, 5, 11, 47, 923, 409 619, 83 763 206 255,. . .
930
the smallest number n such that
φ(n)
n
=
8
31
; the sequence of numbers satis-
fying this equation begins as follows: 930, 1860, 2790, 3720, 4650, 5580, 7440,
8370, 9300, . . .
933
(probably) the largest number which cannot be written as the sum of two
numbers whose index of composition is 3/2: in other words, if 933 = a + b,
then min(λ(a), λ(b)) 3/2; if stands for the largest number n which cannot
be written as n = a + b with min(λ(a), λ(b)) ρ, then we have the following
table (all the values are based on numerical computations and are therefore
only conjectured values)118:
ρ 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
23 23 23 119 933 2 351 8 638 62 471 1 109 549 ? ?
935 (= 5 · 11 · 17)
the second Lucas-Carmichael number (see the number 399);
the smallest square-free composite number n such that p|n =⇒ p + 10|n + 10
(see the number 399).
118In a related matter, V. Blomer [22] established that the number of numbers x which can be
written as the sum of two powerful numbers (thus in particular with an index of composition 2)
is x/(log x)0.253.
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