Those Fascinating Numbers 363
330 645 100 273
the smallest number n such that π(n)
7
j=1
(j 1)!n
logj
n
, this last expression rep-
resenting the first seven terms of the asymptotic expansion of Li(n): here
π(330 645 100 273) = 12 975 900 861 while
∑7
j=1
(j−1)!n
logj n
n=330 645 100 273
12 975 900 860.904 (see the number 73).
403 031 236 608 (=
213
·
32
· 7 · 11 · 13 · 43 · 127)
the
11th
4-perfect number (see the number 30 240).
443 365 544 448
the
17th
powerful number n such that n+1 is also powerful: here 443 365 544 448 =
29
·
32
·
172
·
5772
and 443 365 544 449 =
6658572
(see the number 288).
526 858 348 381
the
13th
horse number (see the number 13).
554 688 278 429
the first term of the longest known Cunningham chain, namely of length 12; a
sequence of prime numbers q1 q2 . . . qk is
called208
a Cunningham chain
of length k if qi = 2qi−1 + 1 for i = 2, 3, . . . , k; in this example, the Cunning-
ham chain is 554688278429, 1109376556859, 2218753113719, 4437506227439,
8875012454879, 17750024909759, 35500049819519, 71000099639039,
142000199278079, 284000398556159, 568000797112319 and 1136001594224639.
608 981 813 029
the smallest number n for which π(n; 3, 1) π(n; 3, 2), where π(n; k, ) stands
for the number of prime numbers p n such that p (mod k) (see C. Bays
& R.H. Hudson [18]).
682 076 806 159
the
18th
Bell number (see the number 52).
682 923 295 390
the
15th
ideal number (see the number 390).
208It is not known if there exist arbitrarily long Cunningham chains.
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