Those Fascinating Numbers 383
2 393 703 338 691 891 312
the smallest known number n for which β∗(n) = β∗(n + 1) = β∗(n + 2) =
β∗(n + 3) = β∗(n + 4) = β∗(n + 5), where β∗(n) = β(n) P (n) =

p|n
pP (n)
p:
indeed,
2 393 703 338 691 891 312 =
24
·
32
· 19 · 37 · 23645718139441,
2 393 703 338 691 891 313 = 61 · 39241038339211333,
2 393 703 338 691 891 314 = 2 · 11 · 17 · 31 · 206460526021381,
2 393 703 338 691 891 315 = 3 · 5 · 53 · 3010947595838857,
2 393 703 338 691 891 316 =
22
· 59 · 10142810757169031,
2 393 703 338 691 891 317 = 7 · 13 · 41 · 641571519349207,
so that the common value of β∗(n + i), for i = 0, 1, 2, 3, 4, 5, is 61 (see the
number 15 860).
3 385 534 663 256 845 323
the 18th horse number (see the number 13).
4 185 296 581 467 695 669
the 100 000 000 000 000
000th
prime number (an estimate due to Marc Del´eglise).
5 317 378 991 792 784 000
the second known 3-powerful number which can be written as the sum of two
co-prime 3-powerful numbers:
373
·
1973
·
3073
=
27
·
34
·
53
·
73
·
22873
+
173
·
1062193
(see the number 776 151 559).
7 156 857 700 403 137 441 (=11 · 13 · 17 · 19 · 29 · 37 · 41 · 43 · 61 · 97 · 109 · 127)
the smallest Carmichael number which is the product of 12 prime numbers (see
the number 41 041).
9 999 999 999 999 999 989
the largest 19 digit prime number.
10 000 000 000 000 000 051
the smallest 20 digit prime number.
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