88 Jean-Marie De Koninck
462
the
12th
number n such that n ·
2n
1 is prime (see the number 115);
the smallest number which cannot be written as the sum of eight non zero
distinct squares (R.K. Guy [101], C20).
463
the first term of the smallest sequence of five consecutive prime numbers all of
the form 4n + 3 (as well as six or seven consecutive prime numbers all of the
form 4n + 3); if we denote by qk the first term of the smallest sequence of k
consecutive prime numbers all of the form 4n + 3, we have the following table:
k qk
1 3
2 3
3 199
4 199
5 463
k qk
6 463
7 463
8 36 551
9 39 607
10 183 091
k qk
11 241 603
12 241 603
13 241 603
14 9 177 431
15 9 177 431
k qk
16 95 949 311
17 105 639 091
18 341 118 307
19 727 334 879
20 727 334 879
(see the number 2 593 for the similar question with 4n + 1);
the prime number which allows one to write the number 6 as the difference
of two powerful numbers: 6 =
54
·
73

4632
= 214 375 214 369, a repre-
sentation discovered by W. Narkiewicz and that S.W. Golomb thought to be
impossible; Mollin & Walsh [141] proved that each integer k 0 has infinitely
many representations as the difference of two powerful numbers.
464
the third solution of σ(n) = 2n + 2: the list of solutions of this equation begins
as follows: 20, 104, 464, 650, 1 952, 130 304, 522 752, 8 382 464,. . .
91
467
the prime number which appears the most often as the tenth prime factor of
an integer (see the number 199).
469
the
15th
number n such that n! 1 is prime (see the number 166).
91It is easy to show that each number n = 2α·p, where α is a positive integer such that p = 2α+1−3
is prime, is a solution of σ(n) = 2n + 2: it is the case when α = 2, 3, 4, 5, 8, 9, 11, 13, 19, 21, 23, 28, 93;
the solutions corresponding to α = 2, 3, 4, 5, 8, 9, 11 are included in the above list.
Previous Page Next Page