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).
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