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, . . .