294 Jean-Marie De Koninck

8 627 527

• the smallest number which can be written as the sum of the cubes of three

distinct prime numbers in three distinct ways: here,

8 627 527 =

193

+

1513

+

1733

=

233

+

1393

+

1813

=

713

+

733

+

1993

(see the number 185 527).

8 870 024

• the smallest number n such that n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8

are all divisible by a square 1: here 8 870 024 = 23 · 1108753, 8 870 025 =

3 · 52 · 227 · 521, 8 870 026 = 2 · 112 · 36653, 8 870 027 = 292 · 53 · 199, 8 870 028 =

22 · 3 · 47 · 15727, 8 870 029 = 72 · 157 · 1153, 8 870 030 = 2 · 5 · 13 · 312 · 71,

8 870 031 =

32

· 311 · 3169, 8 870 032 =

24

· 554377 (see the number 242);

• the smallest number n such that

f(n + 1) = f(n + 2) = f(n + 3) = f(n + 4) = f(n + 5) = f(n + 6) = f(n + 7),

where f(n) stands for the product of the exponents in the factorization of n

(see the number 843).

8 910 720 (=

27

·

32

· 5 · 7 · 13 · 17)

• the smallest solution of

σ(n)

n

=

9

2

; the sequence of numbers satisfying this

equation begins as follows: 8 910 720, 17 428 320, 8 583 644 160,. . .

9 177 431

• the first term of the smallest sequence of 14 consecutive prime numbers each

of the form 4n + 3 (as well as 15 consecutive prime numbers each of the form

4n + 3); see the number 463.

9 511 424

• the smallest number n such that 11! divides 1 + 2 + . . . + n (see the number

224).

9 549 410

• the smallest number which can be written as the sum of the squares of two

prime numbers in 9 (as well as of 10, 11, 12 and 13) distinct ways: 9 549 410 =

2112 + 30832 = 2632 + 30792 = 5032 + 30492 = 5712 + 30372 = 6412 + 30232 =

8572 +29692 = 9912 +29272 = 13012 +28032 = 14272 +27412 = 16372 +26212 =

17472 + 25492 = 18612 + 24672 = 19332 + 24112 (see the number 338).