Those Fascinating Numbers 99
646
the number of Carmichael numbers
109
(Swift, 1975); if nk stands for the
number of
Carmichael103
numbers
10k,
then n4 = 7, n5 = 16, n6 = 43,
n7 = 105, n8 = 255, n9 = 646, n10 = 1 547, n11 = 3 605, n12 = 8 241,
n13 = 19 279, n14 = 44 706, n15 = 105 212 and n16 = 246 683.
649
the smallest solution x of the Fermat-Pell equation
x2

13y2
= 1 (with y = 0):
this solution (x, y) is (649, 180).
650
the fourth solution of σ(n) = 2n + 2 (see the number 464).
651
the smallest number 1 whose sum of divisors is a tenth power: σ(651) = 210.
656
the seventh dihedral perfect number (see the number 130).
657 (=
32
· 73)
the ninth unitary hyperperfect number (see the number 288).
661
the largest known number n such that An := n! (n 1)! + (n 2)! . . . +
(−1)n+11!
is prime: using a computer, one can check that An is prime for n = 3,
4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160 and 661 (see R.K. Guy [101], B43).
666
the third number which is equal to the sum of its digits added to the sum of
the cubes of its digits: the only numbers satisfying this property are 12, 30,
666, 870, 960 and 1 998.
103This is in accordance with the result of W.R. Alford, A. Granville & C. Pomerance [3] according
to which not only the set of Carmichael numbers is infinite, but if C(x) stands for the number of
Carmichael numbers x, then C(x) x2/7 for x sufficiently large; indeed, C(1011) = n11 = 3605
1389.5 =
(1011)2/7.
More recently, G. Harman [108] proved that, for x large enough, C(x)

,
where β 0.33. It is conjectured that, for any ε 0, C(x) x1−ε if x x0(ε).
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