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.