Those Fascinating Numbers 141

2 007

• the sixth number n such that

2n

+

n2

is a prime number: the only numbers

n 30 000 satisfying this

property138

are 3, 9, 15, 21, 33, 2 007, 2 127, 3 759

and 29 355.

2 015

• the third Lucas-Carmichael number (see the number 399).

2 016 (=

25

·

32

· 7)

• the second Erd˝ os-Nicolas number, that is a number n which is not perfect, but

for which there exists a number m such that

∑

d|n, d≤m

d = n; the sequence

of numbers satisfying this property begins as follows: 24, 2 016, 8 190, 42 336,

45 864, 392 448, 714 240, 1 571 328 . . . (with no others 2 · 107); see P. Erd˝ os &

J.L. Nicolas [78];

• the second solution of

σ(n)

n

=

13

4

(see the number 360).

2 041 (= 13 · 157)

• the second 12-hyperperfect number (see the number 697).

2 047

• the smallest composite Mersenne number: 2 047 =

211

− 1 = 23 · 89;

• the smallest strong pseudoprime in base 2: an odd composite number n, with

(a, n) = 1 and n − 1 =

2s

· d, d odd, is said to be a strong pseudoprime in base a

if

ad

≡ 1 (mod n) or if

a2r

d

≡ −1 (mod n) for a certain integer r ∈ [0, s −1].

2 053

• the

11th

known prime number pk such that p1p2 . . . pk − 1 is prime (see the

number 317).

2 056

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

a 16 × 16 magic square (see the number 15).

138It is easy to check that such a number n is necessarily of the form 6k + 3.