124 Jean-Marie De Koninck

1 295

• the smallest solution of τ (n + 1) − τ (n) = 17: the six smallest solutions125 are

1 295, 6 399, 25 599, 117 649, 123 903 and 173 055 (see the number 399).

1 297

• the largest known prime of the form 6n + 1 (here with n = 4);

• the fourth prime number of the form n4 + 1: the smallest ten prime numbers of

this form are 2, 17, 257, 1 297, 65 537, 160 001, 331 777, 614 657, 1 336 337 and

4 477 457; no one knows how to prove that there exist infinitely many prime

numbers of this form (see M. Lal

[121])126,

or even of the form

m2

+ 1.

1 306

• the sixth number n 9 such that n =

∑r

i=1

di,

i

where d1, . . . , dr stand for the

digits of n: here 1 306 =

11

+

32

+

03

+

64

(see the number 175).

1 307

• the smallest number n which allows the sum

m≤n

ω(m)=1

1

m

to exceed 3; the sequence

of the mallest numbers n = n(k) which allow this sum to exceed k begins as

follows: 4, 19, 1 307, 263 215 633, . . . (see the number 277).

1 309

• the smallest number n such that n, n + 1 and n + 2 are square-free and each

have three prime factors: here 1 309 = 7 · 11 · 17, 1 310 = 2 · 5 · 131 and

1 311 = 3 · 19 · 23; the sequence of numbers satisfying this property begins as

follows: 1309, 1885, 2013, 2665, 3729, 5133, 6061, 6213, 6305, 6477, . . . ; if nk

stands for the smallest number n such that n, n + 1 and n + 2 are square-

free, each having k prime

factors127,

then n2 = 33, n3 = 1 309, n4 = 203 433,

n5 = 16 467 033 and n6 = 1 990 586 015.

1 319

• the smallest prime factor of the Mersenne number 2659 − 1.

125Since 17 is odd, it is clear that n or n + 1 must be a perfect square, which allows one to quickly

find the solutions.

126However, recently, J. Friedlander & H. Iwaniec [85] proved that there exist infinitely many prime

numbers of the form

a2

+

b4.

127It is clear that such an integer n satisfies n ≡ 1 (mod 4).