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).
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