122 Jean-Marie De Koninck
1 224
the number of twin prime pairs 105; if π2(x) stands for the number of prime
numbers p with p + 2 prime, we have the following table:
x π2(x)
10 2
102 8
103 35
104 205
105 1 224
106 8 169
107 58 980
x π2(x)
108
440 312
109 3 424 506
1010 27 412 679
1011 224 376 048
1012 1 870 585 220
1013 15 834 664 872
1014 135 780 321 665
1 225
the third triangular number which is also a perfect square: here 1 225 =
49·50
2
=
352
(see the number 36).
1 229
the number of prime numbers 10 000.
1 230
the smallest number n such that the Liouville function λ0 takes successively,
starting with n, the values 1, −1, 1, −1, 1, −1, 1, −1, 1, −1 (see the number 6185).
1 234
the number of digits in the decimal expansion of the Fermat number
2212
+ 1.
1 253
the third number n such that σ(n + 1) = 2σ(n); the sequence of numbers sa-
tisfying this property begins as follows: 5, 125, 1253, 1673, 3127, 5191, 7615,
12035, 43817, 47795, 48559, 49955, 56975, 58373, 61721, 63545, 68033, 78395,
97411, . . . ; if nk stands for the smallest number n such that σ(n + 1) = kσ(n),
then n1 = 206, n2 = 1 253, n3 = 1 919 and n4 = 37 033 919; as for the value
of n5, one may at least say that n5 14 182 439 039, since σ(14182439040) =
5σ(14182439039).
1 257
the fourth solution of σ(φ(n)) = σ(n) (see the number 87).
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