12 Jean-Marie De Koninck
the exponent of the eighth Mersenne prime (2 147 483 647 = 231 1) (Euler,
1750).
32
the smallest solution of φ(n) = φ(n + 28); the sequence of numbers satisfying
this equation begins as follows: 32, 56, 68, 82, 112, 140, 155, 181, 193, 260,
. . . (see the footnote tied to the number 15).
33
the largest number which cannot be written as the sum of five non zero squares:
the others are 1, 2, 3, 4, 6, 7, 9, 10, 12, 15 and 18 (Sierpinski [185], p. 408);
the smallest solution of σ(n) = σ(n + 2); the sequence of numbers satisfying
this equation begins as follows: 33, 54, 284, 366, 834, 848, 918, 1240, 1504,
2910, . . . ;
the smallest number n such that τ (n) = τ (n + 1) = τ (n + 2); the sequence of
numbers satisfying this property begins as follows: 33, 85, 93, 141, 201, 213,
217, 230, 242, 243, 301, 374, 393, 445, 603, 633, 663, 697, 902, 921, . . . ; if nk
stands for the smallest number n such that τ (n) = τ (n+1) = . . . = τ (n+k−1),
then n2 = 14, n3 = 33, n4 = 242, n5 = 11 605, n6 = 28 374, n7 = 171 893 and
n8 = 1 043 710 445 721.
12
34
the smallest solution of σ(n) = σ(n + 19); the sequence of numbers satisfying
this equation begins as follows: 34, 158, 226, 266, 459, 3045, 3518, 3914, 4305,
6236, 8307, . . . ;
the value of the sum of the elements of a diagonal, of a row or of a column of
a 4 × 4 magic square (see the number 15).
35
the smallest solution of φ(x) = 24;
the number of twin prime pairs 1000 (see the number 1 224);
the second solution w of the aligned houses problem: in a village, there is
only one street, all the houses are located on the same side of the street and
are numbered consecutively beginning with the number 1; the maire’s house
12In
1984, Heath-Brown [109] proved that there exists infinitely many numbers n such that τ (n) =
τ (n + 1). So far, no one knows how to prove that there exist infinitely many numbers n such that
τ (n) = τ (n + 1) = τ (n + 2). Interestingly, J.M. De Koninck & F. Luca [58] proved that, for each
integer k 2, one can find infinitely many numbers n such that f (n) = f (n+1) = . . . = f (n+k−1),
where f (n) stands for the product of the exponents in the factorization of n, which is clearly a
function very similar to the τ (n) function.
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