Those Fascinating Numbers 133
1 618
the integer part131 of
ee2
.
1 627
the prime number which appears the most often as the
12th
prime factor of an
integer (see the number 199);
the smallest number n for which π(n)
3
j=1
(j 1)!n
logj
n
, this last expression
representing the first three terms of the asymptotic expansion of Li(n): here
π(1627) = 258 while
n
log n
+
n
log2 n
+
2n
log3 n
n=1627
257.832 (see the number
73);
the ninth prime number p such that
(3p
1)/2 is itself a prime number (see
the number 1 091).
1 634
the second number which can be written as the sum of the fourth powers of its
digits: 1 634 =
14
+
64
+
34
+
44;
the others are 1, 8 208 and 9 474;
the sixth solution of σ(n) = σ(n + 1) (see the number 206).
1 638
the fourth number which is neither perfect nor multi-perfect but whose har-
monic mean is an integer (see the number 140);
the fourth solution of
σ(n)
n
=
8
3
(see the number 1 488).
1 639
the 1
000th
square-free number (see the number 165).
131The
interest for this number stems from a number theory result according to which the
kth
prime factor qk (n) of a number n is “usually” of the order of
eek
, in the sense that, for all ε 0
and any function ξ(n) which tends to +∞ as n ∞,
sup
ξ(n)≤k≤ω(n)
log log qk(n) k
2k log k
1 + ε
almost everywhere. This result was first stated by P. Erd˝ os in 1946 (see G. Tenenbaum [193], p. 344).
Thus, q2 1618, q3 528 491 311 and q4 514 843 556 263 450 564 886 528.
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