102 Jean-Marie De Koninck
k N3(10k)
1 2
2 7
3 20
4 51
5 129
6 307
7 813
8 1645
9 3721
10 8348
k N3(10k)
11 18589
12 41136
13 90619
14 198767
15 434572
16 947753
17 2062437
18 4480253
19 9718457
20 21055958
k N3(10k)
21 45575049
22 98566055
23 213028539
24 460160083
25 993533517
26 2144335391
27 4626664451
28 9980028172
29 21523027285
30 46408635232
714
the largest known number n such that n(n + 1) = p1p2 . . . pk for a certain k:
here 714 · 715 = 2 · 3 · 5 · 7 · 11 · 13 · 17;
the ninth number n such that β(n) = β(n + 1); the sequence of numbers satis-
fying this property begins as follows: 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682,
2107, 2299, 2600, 2783, 5405, 6556, 6811, 8855, 9800, . . . ; the numbers 714 and
715 are often called the Ruth-Aaron numbers because on April 8, 1974, Hank
Aaron broke the home run record of 714 set by the famous Babe Ruth by hitting
his own 715th home run; no one has yet proved that equation β(n) = β(n + 1)
has infinitely many solutions (see Pomerance [164]);
the smallest number n such that β(n) β(n + 1) . . . β(n + 5), where
β(n) =

p|n
p: here 29 29 181 242 361 719; if nk stands for the
smallest number n such that β(n) β(n + 1) . . . β(n + k 1), then we
have the following table:
k 3 4 5 6 7 8
nk 4 4 90 714 9 352 16 575
k 9 10 11 12
nk 617 139 721 970 6 449 639 1 303 324 906
(compare with the table given at number 46 189);
the fifth number n such that φ(n)σ(n) is a fourth power: φ(714)σ(714) = 244
(see the number 170).
715 (= 5 · 11 · 13)
the second number n such that
2n
7 is a prime number (see the number 39);
the smallest square-free composite number n such that p|n =⇒ p + 5|n + 5 (see
the number 399).
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