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).