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. 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. 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).
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