150 Jean-Marie De Koninck

2 657

• the tenth prime number pk such that p1p2 . . . pk + 1 is prime (see the number

379).

2 667

• the smallest number larger than 1 and whose sum of divisors is a sixth power:

σ(2667) = 46.

2 673

• the smallest number which can be written as the sum of three fourth powers in

two distinct ways: 2 673 =

24

+

44

+

74

=

34

+

64

+

64

(see the number 6 578).

2 685

• the seventh solution of σ(n) = σ(n + 1) (see the number 206).

2 687

• the smallest prime factor of the Mersenne number

279

− 1, whose complete

factorization is given by

279

− 1 = 2687 · 202029703 · 1113491139767.

2 693

• the smallest prime number p such that 12p−1 ≡ 1 (mod p2): the only other

prime number p 232 satisfying this congruence is 123 653 (see Ribenboim

[169], p. 347).

2 699

• the rank of the prime number which appears the most often as the

18th

prime

factor of an integer: p2699 = 24 251 (see the number 199).

2 719

• the largest odd number n 150 000 which cannot be written in the form

x2

+

y2

+

10z2

(see H.G. Gupta [100]).

2 728 (=

23

· 11 · 31)

• the second number n such that β(n)|n, β(n + 1)|(n + 1), β(n + 2)|(n + 2) and

β(n + 3)|(n + 3): indeed, here β(2728) = 44, β(2729) = 2729, β(2730) = 30

and β(2731) = 2731; the sequence of numbers satisfying this property begins

as follows: 29, 2728, 3526, 103966, 150587, 4743197, . . .