Those Fascinating Numbers 37

k n = nk insolite

1 111

2 11 112

3 1 111 211 136

4 11 121 114 112

5 11 . . . 1

11

3111113 11 . . . 1

11

75 (*)

k n = nk insolite

6 122 121 216

7 123 412 112 474 112

8 121 111 216 128

9 911 131 213 824

in this table, the star (*) next to the number n5 indicates that it is the smallest

known insolite number with this property;

• the value of the sum of the elements of a diagonal, of a line or of a column in

a 6 × 6 magic square (see the number 15).

112 (=

24

· 7)

• the smallest number n having at least two distinct prime factors and which is

such that p|n =⇒ p + 8|n + 8; the sequence of numbers satisfying this property

begins as follows: 112, 135, 432, 532, 832, 847, 1372, 1792, 2632, 3072, 8092,

8722, . . .

113

• the number n at which the quotient Q(n) =

π(n)

n/ log n

reaches its maximal value

(see Rosser & Schoenfeld [178]); in fact Q(113) ≈ 1.25506; moreover, it is the

only number n ≥ 2 for which the inequality π(n)

5

4

n

log n

does not hold (since

30 29.8791);

• the prime number which appears the most often as the seventh prime factor of

an integer (see the number 199).

114

• the smallest solution of σ2(n) = σ2(n + 19).

115

• the seventh number n such that

n·2n

−1 is prime: the only numbers n 700 000

satisfying this property, also at times called Woodall numbers, are 2, 3, 6, 30,

75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 882, 5 312, 7 755, 9 531, 12 379,

15 822, 18 885, 22 971, 23 005, 98 726, 143 018, 151 023 and 667 071 (a result due

to Keller).