Those Fascinating Numbers 13

is located in such a manner that the sum of the numbers of the houses to

its left is equal to the sum of the numbers to its right: the number on the

maire’s house is therefore the solution of equation 1 + 2 + . . . + (w − 1) =

(w + 1) + (w + 2) + . . . + (w + s): the smallest solution is given by w = 6

and w + s = 8, while the next six are w = 35 and w + s = 49, w = 204 and

w + s = 288, w = 1189 and w + s = 1681, w = 6930 and w + s = 9800,

w = 40391 and w + s = 57121, w = 235416 and w + s = 332928; it is possible

to prove that the aligned houses problem has infinitely many solutions; on the

other hand, it is interesting to observe that there exists a connection between

this problem and the numbers n for which n and n + 1 are

powerful13.

36

• the smallest triangular number 1 which is also a perfect square:

8(8+1)

2

= 62;

the sequence of numbers satisfying this property begins as follows: 1, 36, 1225,

41616, 1413721, 48024900, 55420693056, . . . ; there exist infinitely many such

numbers14;

• the largest solution n of equation

d|n

τ (d) = n: the only solutions of this equa-

tion are 1, 3, 18 and 36.

37

• the median value of the second prime factor of a number: indeed, one can show

that the probability that the second prime factor of a number is ≤ 37 is equal

to 0.500248 . . . ≈

1

2

; the median value of the third prime factor of a number is

42 719, while

that15

of the fourth one is 5 737 850 066 077;

• the smallest irregular prime number (see the number 59);

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

as x1

5

+ x2

5

+ . . . + xr

5,

where the xi’s are non negative integers (Chen, 1964):

see the number 4;

13Indeed,

since

1 + 2 + . . . + (w − 1) = (w + 1) + (w + 2) + . . . + (w + s),

it is easy to see that for any other solution (w, s) of the aligned houses problem, we have

(∗) (w + s)(w + s + 1) =

2w2.

This is why, since (w + s, w + s + 1) = 1, it follows from (∗) that if w is odd, the numbers w + s and

w + s + 1 must both be powerful. This explains why we find the numbers 288, 9800 and 332 928

amongst the numbers n for which n and n + 1 are powerful (see the number 288).

14Indeed,

this follows from the fact that the diophantine equation n(n + 1) =

2m2

has a solution

m if and only if m appears in the sequence (an)n≥0 defined by a0 = 1, a1 = 6 and, for n ≥ 2, by

an = 6an−1 − an−2.

15In 2002, J.M. De Koninck & G. Tenenbaum [63] proved that if p∗

k

stands for the median value

of

kth

prime factor of an integer, then log log

p∗

k

= k − b + O(1/

√

k), where b =

1

3

+ γ − ∑the

p

log

(

1

1−1/p

)

−

1

p

≈ 0.59483 (here γ ≈ 0.557215 stands for Euler’s constant); on the other

hand, they proved that if the Riemann Hypothesis is true, then p∗

5

≈ 7.887 · 1034.