110 Jean-Marie De Koninck

n6 = 594 839 010 = 2 · 3 · 5 · 17 · 29 · 37 · 1087 =

26

+

56

+

296,

n7 = 275 223 438 741 = 3 · 23 · 43 · 92761523 =

37

+

237

+

437,

n8 = 26 584 448 904 822 018 = 2 · 3 · 7 · 17 · 19 · 113 · 912733109

=

28

+

178

+

1138,

n9 = 40 373 802 = 2 ·

34

· 7 · 35603 =

29

+

39

+

79,

n10 = 420 707 243 066 850 = 2 ·

32

·

52

· 29 · 32238102917

=

210

+

510

+

2910;

• the fourth number which is equal to the sum of its digits added to the sum of

the cubes of its digits: one can easily prove that the only numbers satisfying

this property are 12, 30, 666, 870, 960 and 1 998.

871

• the smallest number n which allows the sum

m≤n

Ω(m)=2

1

m

to exceed 2; if, given

k ≥ 1, nk stands for the smallest number n which allows this sum to exceed k,

then the sequence (nk)k≥1 begins as follows: 35, 871, 43 217, 5 296 623, . . .

872

• the

16th

number n such that n! + 1 is prime (see the number 116).

873

• the value of 1! + 2! + 3! + 4! + 5! + 6!.

877

• the seventh Bell number (see the number 52).

880

• the number of 4 × 4 magic squares (excluding rotations and reflections); if nk

stands for the number of k × k magic squares, then n1 = 1, n2 = 0, n3 = 1,

n4 = 880 and n5 = 275 305 224: this last value was obtained by R. Schroeppel

in 1973.

881

• the largest known prime number p such that P

(p2

−1) = 11, where P (n) stands

for the largest prime factor of n (see the number 4 801).