70 Jean-Marie De Koninck
296
the smallest number n which allows the sum
m≤n
1
φ(m)
to exceed 11 (see the
number 177).
299
the fifth number m such that

n≤m
σ(n) is a perfect square: here

σ(n) =
73 441 =
2712;
if we denote by mk the
kth
number such that sk =
∑≤299n
n≤mk
σ(n)
is a perfect square, here are the values of mk, for 1 k 10, with the corres-
ponding values of sk:
k mk sk

sk
1 1 1 1
2 2 4 2
3 53 2304 48
4 174 24964 158
5 299 73441 271
k mk sk

sk
6 1377 1560001 1249
7 12695 132549169 11513
8 44469 1626428241 40329
9 423922 147805647025 384455
10 2068248 3518227981636 1875694
300 (= 22 · 3 · 52)
the smallest number with at least two distinct prime factors and which is di-
visible by the square of the sum of its prime factors: the sequence of numbers
satisfying this property begins as follows: 300, 600, 900, 980, 1008, 1200, 1500,
1575, 1800, 1960, . . .
301
the smallest 6-hyperperfect number: we say that a number n is 6-hyperperfect
if it can be written as n = 1 + 6
d|n
1dn
d (which is equivalent to the condition
6σ(n) = 7n + 5); the smallest four 6-hyperperfect numbers are 301, 16 513,
60 110 701 and 1 977 225 901; the number 2 733 834 545 701 is also 6-hyperperfect
and perhaps the fifth one (see the number 21).
303
the ninth number k such that
k|(10k+1
1); the sequence of numbers satisfying
this property begins as follows: 1, 3, 9, 11, 33, 77, 99, 143, 303, 369, 407, 707,
959, 1001, 1111, . . .
307
the third number which is not a palindrome, but whose square is a palindrome
(see the number 26).
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