374 Jean-Marie De Koninck
119 429 556 097 859
the eighth number (and the largest one known) which can be written as the
sum of the cubes of its prime factors:
119 429 556 097 859 = 7 · 53 · 937 · 6983 · 49199
=
73
+
533
+
9373
+
69833
+
491993
(see the number 378).
130 429 015 516 800 (= 27 · 33 · 52 · 72 · 11 · 13 · 17 · 19 · 23 · 29)
the smallest number n such that σ(n) 6n: here σ(n)/n = 6.017 . . . (see the
number 27 720).
170 824 677 031 250
possibly the number n at which the quantity Q4(n) := min(λ(n), λ(n+1), λ(n+
2), λ(n +3)) reaches its maximal value, namely approximately 1.41419: indeed,
we have
Q4(170824677031250) min(1.41776, 1.41419, 1.44547, 1.42225)
= 1.41419;
if, for each number k 2, nk stands for the number n at which the quantity
Qk(n) := min(λ(n), λ(n + 1), . . . , λ(n + k 1))
reaches its maximal value, the following table gives the conjectured values of
nk for 2 k 7:
k nk Qk(n)
2 63 927 525 375 2.65551
3 85 016 574 1.72085
4 170 824 677 031 250 1.41419
5 82 564 350 1.28177
6 51 767 910 1.22783
7 244 294 249 389 674 423 328 124 1.66671
no number n for which Q8(n) 8/7 is known, although common sense indicates
that there are infinitely many such numbers (see the footnote tied to the number
14 018 750).
171 160 044 505 600 (=
211
·
34
·
52
·
72
· 11 · 13 · 19 · 31)
the smallest solution of
σ(n)
n
=
11
2
.
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