140 Jean-Marie De Koninck

1 935

• the smallest solution of σ2(n) = σ2(n + 18).

1 941

• the smallest number n such that

∑

m≤n

τ (m) is a multiple of 1 000 (here the

sum is equal to 15 000).

1 952

• the fifth solution of σ(n) = 2n + 2 (see the number 464).

1 953 (=

32

· 7 · 31)

• the smallest number n which allows the sum

m≤n

ω(m)=3

1

m

to exceed 1; the sequence

of the smallest numbers n = n(k) which allow this sum to exceed k begins as

follows: 1 953, 26 277, 346 065, 5 099 011, . . .

1 963

• the

18th

number n such that n! − 1 is prime (see the number 166).

1 983 (=

32

· 7 · 31)

• the third number n such that

2n

− 7 is prime (see the number 39).

1 998

• the sixth and largest number which is equal to the sum of its digits added to

the sum of the cubes of its digits: the only numbers with this property are 12,

30, 666, 870, 960 and 1 998.

1 999

• the second prime number p such that ω(p+1) = 2, ω(p+2) = 3 and ω(p+3) = 4

(see the number 103);

• the largest number n such that f3(n) n, where f3(n) = f3([d1, d2, . . . , dr]) =

d1

3

+ d2

3

+ . . . + dr

3,

with d1, d2, . . . , dr standing for the digits of n.