Those Fascinating Numbers 21
the smallest prime number p such that Ω(p + 1) = 2 and Ω(p + 2) = 3; if for
each positive integer k, we let qk be the smallest prime number q such that
Ω(q + 1) = 2, Ω(q + 2) = 3, . . . , Ω(q + k) = k + 1, then q1 = 3, q2 = 61,
q3 = 193, q4 = 15 121, q5 = 838 561 and q6 = 807 905 281;
the smallest number n such that τ (n) τ (n + 1) τ (n + 2); it is also the
smallest number n such that τ (n) τ (n + 1) τ (n + 2) τ (n + 3): here
2 4 6 7 (see the numbers 11 371 and 7 392 171).
62
the second solution of σ(n) = σ(n + 7); the sequence of numbers satisfying this
equation begins as follows: 10, 62, 188, 362, 759, 1178, 1214, 1431, 1442, 1598,
1695, 1748, 2235, 3495, 6699, 9338, 9945, . . . ;
the second solution of σ(n) = σ(n + 15) (see the number 26).
63
the value of the Kaprekar constant for the two digit numbers (see the number
495).
64
the smallest number n such that ω(n) = 1, ω(n +1) = 2 and ω(n +2) = 3: here
64 = 26, 65 = 5 · 13 and 66 = 2 · 3 · 11; if we denote by nk the smallest number
n such that ω(n) = 1, ω(n + 1) = 2, . . . , ω(n + k) = k + 1, then n2 = 64,
n3 = 1 867, n4 = 491 851 and n5 = 17 681 491.
65
the smallest square-free number which can be written as the sum of two squares
in two distinct ways: here 65 =
82
+
12
=
72
+
42;
it is therefore the smallest
hypotenuse common to two Pythagorean triangles;
the value of the sum of the elements of a diagonal, of a line or of a column of
a 5 × 5 magic square (see the number 15);
the number of possible (straight) paths one can use to move from a given point
to another inside a set E made up of six points in the cartesian plan without
going through each point of E more than
once29.
29In
the general case, if E contains n points, the corresponding number N of possible paths is
given by N =
∑n−2
i=0
i!
(
n−2
i
)
= (n 2)!
∑n−2
i=0
1
i!
e(n 2)!; hence, for n 3, the first terms of
the sequence are 2, 5, 16, 65, 326, 1957, . . .
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