38 Jean-Marie De Koninck

116

• the tenth number n such that n!+1 is prime: the only known numbers satisfying

this property are 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427,

872, 1477, 6 380 and 26 951.

117

• the tenth number n such that n ·

10n

− 1 is

prime45;

the sequence of numbers

satisfying this property begins as follows: 2, 3, 8, 11, 15, 39, 60, 72, 77, 117,

183, 252, 396, 1745, 2843, . . . (see the number 363).

118

• the smallest possible sum common to four triplets of numbers having same

sum and same product, namely the triplets (14,50,54), (15,40,63), (18,30,70)

and (21,25,72): (Problem E2872, Amer. Math. Monthly 89 (1982), p. 499);

• the largest number which cannot be written as the sum of three powerful num-

bers (=

1).46

119

• the largest solution n of the diophantine equation

1

2

n(n+1) =

1

6

m(m+1)(m+2):

the solutions (n, m) of this equation are (1,1), (4,3), (15,8), (55,20) and (119,

34): see the number 10;

• the eighth number 1 whose sum of divisors is a perfect square: σ(119) = 122;

• the third number n such that φ(n)σ(n) is a cube: the sequence of numbers sa-

tisfying this property begins as follows: 1, 3, 119, 357, 2522, 6305, 6596, 6604,

7566, 18915, 19788, 19812, 20520,. . . (see the number 170);

• the smallest number n satisfying φ(n) = 3φ(n + 1); the sequence of numbers

satisfying this equation begins as follows: 119, 527, 545, 2849, 3689, 4487, 6649,

18619, 26771, 30377, 44659, 47585, 50507, 76997, 83021, . . . (see the number

629);

• (probably) the largest number which cannot be written as the sum of two co-

prime numbers each having an index of composition ≥ 1.4 (see the number

933).

45It

is easy to prove that if n ≡ 1 (mod 3), then the corresponding number n ·

10n

− 1 is not

prime since it is a multiple of 3; see the number 363 for the idea behind the proof.

46Since

any number which is not of the form 4 (8k + 7), with ≥ 0 and k ≥ 1, can be written

as the sum of three squares (see Grosswald [99]), it is clear that one only needs to verify that if n

is of the form n = 8k + 7, then it can be written as the sum of three powerful numbers. The only

numbers 8k + 7 127 which can be written as the sum of three powerful numbers are 39, 47, 55,

63, 71, 79, 95 and 103. Moreover, if κ(n) stands for the number of representations of n as the sum

of three powerful numbers, then one can prove that limn→∞ κ(n) = +∞. Finally, if nj stands for

the largest number n having exactly j representations as the sum of three powerful numbers, then

n1 = 399, n2 = 1 263, n3 = 1 335, n4 = 2 103 and n5 = 1 991.