298 Jean-Marie De Koninck
12 792 675
the sixth number n such that β(n)|β(n + 1) and β(n + 1)|β(n + 2): here 74|222
and 222|412698 (see the number 225 504).
12 870 973
the smallest number n such that π(n)
5
j=1
(j 1)!n
logj
n
, this last expression
representing the first five terms of the asymptotic expansion of Li(n): here
π(12870973) = 841464 while
∑5
j=1
(j−1)!n
logj n
n=12870973
841463.3 (see the
number 73).
12 999 168 (=
29
·
32
· 7 · 13 · 31)
the fourth solution of
σ(n)
n
=
11
3
(see the number 35 640).
13 053 769
the smallest perfect square m5 2 for which there exist numbers m1, m2, m3 and
m4 such that mi 2 −(mi−1)2 = mi−1 2 for i = 2, 3, 4, 5: here 13 053 769 = 36132 =
36122 + 852 = 36122 + 842 + 132 = 36122 + 842 + 122 + 52 = 36122 + 842 +
122 + 42 + 32.
13 141 793
the smallest number n such that σ(n), σ(n + 1), . . . , σ(n + 5) all have the same
prime factors, namely here the primes 2, 3, 5, 7 and 149 (see the number 3 777).
13 466 917
the exponent of the
39th
Mersenne prime
213 466 917
1 discovered by the Cana-
dian Michael Cameron in November 2001 using the programme developed by
G. Woltman (see the number 1 398 269).
13 562 027
the smallest number n which allows the sum
i≤n
1
i
to exceed 17 (see the number
83).
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