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.
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