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).
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