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].