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).