236 Jean-Marie De Koninck
103 511
the 10
000th
prime power, in fact here a prime (see the number 419).
104 729
the 10 000th prime number (see the number 541).
105 212
the number of Carmichael numbers
1015
(see the number 646).
106 755
the smallest six digit Sastry number (see the number 6 099).
106 853
the smallest prime number q such that

p≤q
p is a multiple of 10 000: here

p≤106 853
p = 515 530 000 (see the number 35 677).
109 297
the eighth number k such that 11 . . . 1
k
is prime (discovered by Bourdelais and
Dubner in 2007); see the number 19.
109 306
the third number which is not a cube, but which can be written as the sum of
the cubes of some of its prime factors: here 109 306 = 2·31·41·43 =
23+313+433
(see the number 378, as well as 870); besides the numbers which can be written
as the sum of the cubes of their prime factors (see the number 378 for a list
of eight such numbers) and besides the number 109 306, the following numbers
also satisfy this
property172:
23 391 460 =
22
· 5 · 23 · 211 · 241 =
23
+
2113
+
2413,
173 871 316 =
22
· 223 · 421 · 463 =
23
+
4213
+
4633,
450 843 455 098 = 2 · 6007 · 6089 · 6163 =
23
+
60073
+
61633.
172It is easy to prove that if Hypothesis H is true, then there exist infinitely many such numbers.
Indeed, this follows from the fact that under this hypothesis, there exist infinitely many even numbers
k such that the two numbers r = k2 9k + 21 and p = k2 7k + 13 are prime, in which case by
setting n = 2rqp, where q =
k2
8k + 21, it is easy to prove that n =
23
+
r3
+
p3.
This result as
well as other results on this topic can be found in De Koninck & Luca [54].
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