Those Fascinating Numbers 61
n7 = 965 009 045; on the other hand, one can show71 that n8 5 708 691 485,
while n8 n = 65 893 166 030 = 2 · 5 · 17 · 19 · 23 · 43 · 47 · 277 since n + 1 =
3 · 7 · 11 · 29 · 31 · 41 · 71 · 109 (see the number 135 for the analogue problem with
the Ω(n) function).
231 (= 3 · 7 · 11)
the smallest square-free composite number n such that p|n =⇒ p + 9|n + 9 (see
the number 399).
233
the sixth prime Fibonacci number (see the number 89);
the smallest prime factor of the Mersenne number 229 1, whose complete
factorization is given by
229
1 = 536 870 911 = 233 · 1103 · 2089;
the number
229
−1 is the smallest Mersenne number having three distinct prime
factors (see the number 223).
234
the largest solution n 109 of
σ(n)
n
=
7
3
: the smallest one is n = 12.
235
the fourth three digit number = 100, 200, 300, and whose square contains no
more than two distinct digits:
2352
= 55 225 (see the number 109).
236
the number n which allows the sum
m≤n
ω(m)=2
1
m
to exceed 2 (see the number 44).
237
the fourth number n such that
2n−1
+ n is a prime number: the only numbers
n 40 000 satisfying this property are 1, 3, 7, 237 and 1 885.
71It is clear that, for each number k 1, (nk + 1)2 nk(nk + 1) p1p2 . . . p2k, so that
nk

p1p2 . . . p2k, from which it follows that n8 5 708 691 485, and moreover that the inequality
is strict once we verify that the number 5 708 691 485 does not have the required property.
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