Those Fascinating Numbers 57

209

• the ninth number n for which the distance from

en

to the nearest integer is the

smallest (see the number 178).

210

• the largest number n for which Q2(n) = π(n) − π(n/2), where Q2(n) stands

for the number of representations of n as the sum of two odd prime numbers

(see J.M. Deshouillers, A. Granville, W. Narkiewicz & C. Pomerance [64]): here

Q2(210) = 19 = π(210) − π(105) = 46 − 27;

• the area common to the two smallest Pythagorean triangles with same area,

but distinct hypotenuses: these are the triangles (21, 20, 29) and (35, 12, 37);

see Sierpinski [185], p. 46;

• the smallest number 1 which is both triangular and pentagonal:

210 =

20 · 21

2

=

12(3 · 12 − 1)

2

: a number is said to be pentagonal if it is of the

form

k(3k−1)

2

: the sequence of numbers which are both triangular and pentag-

onal begins as follows: 210, 40 755, 7 906 276, 1 533 776 805, . . .

211

• the smallest prime number which is equally distant, by a distance of 12, from the

preceding and following prime numbers: p46 = 199, p47 = 211 and p48 = 223;

• the fourth Euclid number; the sequence (nk)k≥1 of Euclid numbers is defined

by nk = 1+p1p2 . . . pk and begins as follows: 3, 7, 31, 211, 2311, 30031, 510511,

9699691, 223092871, 6469693231, . . . ; the numbers nk are prime at least for

k = 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616 and 643, that is for the

corresponding prime numbers pk = 2, 3, 5, 7, 11, 31, 379, 1 019, 1 021, 2 657,

3 229, 4 547 and 4 787; it is not known if the sequence (nk)k≥1 contains infinitely

many primes.

212

• the third three digit number = 100, 200, 300, and whose square contains no

more than two distinct digits: 2122 = 44 944 (see the number 109).

213

• the number of Niven numbers ≤ 1000; if N(x) stands for the number of Niven

numbers ≤ x, then we have the following

table69:

69In

2003, J.M. De Koninck, N. Doyon & I. K´ atai [50] proved that N(x) = (1 + o(1))cx/ log x,

where c =

14

27

log 10 ≈ 1.19393. In 2005, C. Mauduit, C. Pomerance & A. S´ ark¨ ozy [133] improved

this estimate by providing an error term, more precisely by establishing that N(x) = cx/ log x +

O(x/

log9/8

x).