120 Jean-Marie De Koninck

1 097

• the smallest number n such that φ10(n) = 2, where φ10(n) stands for the tenth

iteration of the φ function (see the number 137).

1 104

• the ninth Keith number (see the number 197).

1 105 (= 5 · 13 · 17)

• the second Carmichael number (see the number 561);

• the smallest number which can be written as the sum of two squares in four

distinct ways, namely 1 105 = 42 + 332 = 92 + 322 = 122 + 312 = 232 + 242 (see

the number 50);

• the value of the sum of the elements of a diagonal, of a line or of a column in

a 13 × 13 magic square (see the number 15).

1 111

• the 15th number k such that k|(10k+1 − 1) (see the number 303).

1 151

• the smallest prime number that is preceded by 21 consecutive composite num-

bers; indeed, there are no prime numbers between 1 129 and 1 151.

1 156 (= 342)

• the smallest number which can be written as the sum of k perfect squares

for each positive integer k ≤ 1000 (Sierpinski [185], p. 410); an example of a

representation of 342 as the sum of 1000 squares is 342 = 2 · 82 +2 · 42 +996 · 12.

1 167

• the largest number which cannot be written as the sum of five composite num-

bers (R.K. Guy [101], C20).

1 184

• the number which, when paired with the number 1 210, forms an amicable pair;

this pair was discovered by Paganini when he was only 16 years old (see the

number 220).