308 Jean-Marie De Koninck

32 509 439

• the tenth solution of σ2(n) = σ2(n + 2) (see the number 1 079).

32 535 999

• the smallest number n such that n and n+1 each have 12 prime factors counting

their multiplicity: here 32 535 999 =

310

· 19 · 29 and 32 536 000 =

26

·

53

·

72

· 83

(see the number 135).

32 582 657

• the exponent of the

44th

Mersenne prime

232 582 657−1

(a 9 808 358 digit number)

discovered by Curtis Cooper and Steven Boone on September 4, 2006, using

the programme developed by G. Woltman (see the number 1 398 269).

32 694 619

• the smallest prime number q1 such that each number qi = 3qi−1 + 2 is prime

for i = 2, 3, . . . , 8 (see the number 575 119): such a sequence of prime numbers

is somewhat similar to the Cunningham chains (see the number 1 122 659).

33 550 336 (= 212(213 − 1))

• the fifth perfect number (the smallest four are 6, 28, 496 and 8 128).

33 721 216

• the

12th

dihedral perfect number (see the number 130).

33 757 004

• the number n which allows the sum

m≤n

ω(m)=2

1

m

to exceed 6 (see the number 44).

34 012 224

• the smallest number n 1 whose sum of digits is equal to

6

√

n: the only other

numbers n 1 satisfying this property are 8 303 765 625, 24 794 911 296 and

68 719 476 736;

• the smallest number which can be written as the sum of three distinct cubes

in ten distinct ways:

34 012 224 =

363

+

2163

+

2883

=

393

+

1533

+

3123

=

413

+

1143

+

3193

=

453

+

2463

+

2673

=

723

+

1143

+

3183

=

1003

+

1923

+

2963

=

1023

+

2273

+

2773

=

1183

+

1863

+

2963

=

1623

+

2163

+

2703

=

1733

+

2143

+

2673

(see the number 1 009).