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.
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