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).
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