334 Jean-Marie De Koninck

731 601 000

• the eighth number n such that φ(n) + σ(n) = 4n (see the number 23 760).

740 461 601

• the smallest number n which allows the sum

i≤n

1

i

to exceed 21 (see the number

83).

745 988 807

• the smallest prime factor of the Mersenne number 2109 − 1, whose complete

factorization is given by

2109

− 1 = 745 988 807 · 870 035 986 098 720 987 332 873.

746 444 160 (= 27 · 34 · 5 · 7 · 112 · 17)

• the smallest solution of

σ(n)

n

=

19

4

.

776 151 559 (=

9193)

• the smallest 3-powerful number which can be written as the sum of two co-

prime 3-powerful numbers: here

9193

=

2713

+

23

·

35

·

733;

A. Nitaj [153]

proved a conjecture of Erd˝ os going back to 1975 which claims that there exist

infinitely

many199

triplets of 3-powerful numbers a, b, c, with (a, b) = 1, such

that a + b = c; this by the way reveals a second 3-powerful number which can

be written as the sum of two 3-powerful numbers, namely the number

11205183603973252067=373

·

1973

·

3073 =27

·

34

·

53

·

73

·

22873

+

173

·

1062193;

however, there exist other 3-powerful numbers with such a representation, such

as the number 3518958160000, for which

3518958160000 =

27

·

54

·

3533

=

34

·

293

·

893

+

73

·

113

·

1673.

199Indeed,

this comes from the fact that equation (∗)

x3

+

y3

=

6z3

has infinitely many solutions.

To prove this, first observe that one can generate infinitely many solutions (xk, yk, zk) of (∗) by

considering the sequence defined by x0 = 37, y0 = 17, z0 = 21 and for each k ≥ 0, by

xk+1 = xk(xk

3

+ 2yk),

3

yk+1 = −yk(2xk

3

+ yk),

3

zk+1 = zk(xk

3

−

yk).3

In order to show that the numbers a = x3

k

, b = y3

k

and c = 6z3

k

are adequate, that is 3-powerful for

each k ≥ 1, it remains to prove that zk is divisible by 6 for each integer k ≥ 1. But this is trivial.

Indeed, on the one hand, since xk and yk are odd for each k ≥ 0, it follows that zk is even for each

k ≥ 1. On the other hand, since 3|z0, then 3|z1, implying that all the other zi’s are also multiples

of 3.