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).