Those Fascinating Numbers 45

an integer: the harmonic mean of a number n, denoted by H(n), is defined by

H(n) = τ (n) ×

⎛

⎝

d|n

1

d

⎞−1

⎠

=

nτ (n)

σ(n)

;

the sequence of numbers satisfying this property begins as follows: 1, 140, 270,

1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400,

173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480,

950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800,

4358600, 4713984, 4754880, 5772200, 6051500, 8506400, 8872200, . . . ;

• the largest number which when raised to the square becomes a tetrahedral

number (Sierpinski [185], p. 87): the only numbers satisfying his property are

1, 2 and 140.

141

• the smallest number n 1 such that n ·

2n

+ 1 is prime; numbers of the form

n ·

2n

+ 1 are called Cullen numbers

56;

it is not known if infinitely many of

these numbers are prime: the only known prime numbers of the form n ·

2n

+ 1

are those with n equal to 141, 4 713, 5 795, 6 611, 18 496, 32 292, 32 469, 59 656,

90 825, 262 419, 361 275 and 481 899 (see W. Keller [118]).

142

• the smallest solution of σ(n) = σ(n + 17); the sequence of numbers satisfying

this equation begins as follows: 142, 238, 418, 429, 598, 622, 2985, 3502, . . . ;

• the only solution y of the diophantine equation x(x+1)(x+2)(x+3)(x+4)(x+

5) =

y2

− 4, the solution (x, y) being (3, 142) (L.E. Mattics

[132])57.

143

• the smallest number r which has the property that each number can be written

as x1 7 +x2 7 +. . .+xr, 7 where the xi’s are non negative integers (see the number 4);

• the number of three digit prime numbers (see the number 21).

56Father Cullen first considered these numbers in 1905.

57Mattics

studied the equation x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5) =

y2

− k, for |k|≤ 31, and

obtained that this diophantine equation has solutions for k = 4 (with (x, y) = (3, 142)), k = 9

(with (x, y) = (1, 27)) and k = 25 (with (x, y) = (21, 12875)); he also proved that for the positive

solutions, we must have 1 ≤ x ≤ max(27, (40|k| + 1)1/3).