328 Jean-Marie De Koninck
299 980 800 (= 211 · 33 · 52 · 7 · 31)
the third solution of
σ(n)
n
=
13
3
(see the number 18 506 880).
305 635 357 (= 43 · 1039 · 6841)
the smallest composite number n such that σ(n + 4) = σ(n) + 4: indeed, since
305 635 357 = 43 · 1039 · 6841, we have
σ(305635357) = σ(43) · σ(1039) · σ(6841) = 44 · 1040 · 6842 = 313089920
and since 305635361 = 41 · 7454521, we have
σ(305635361) = σ(41) · σ(7454521) = 42 · 7454522 = 313089924;
hence, σ(305635357 + 4) = σ(305635357) + 4 (see also the number 434).
315 746 063
the third prime number p such that
83p−1
1 (mod
p2)
(see the number
4 871).
321 197 185 (= 5 · 19 · 23 · 29 · 37 · 137)
the smallest Carmichael number which is the product of six prime numbers (see
the number 41 041).
333 333 331 (= 17 · 19 607 843)
the smallest composite number of the form
1
3
(10k
7): here k = 9 (an observa-
tion due to A. Makowski); the numbers k 4 000 for which
1
3
(10k
7) is prime
are k = 2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, 784, 1 732 and
1 918.
341 118 307
the first term of the smallest sequence of 18 consecutive prime numbers all of
the form 4n + 3 (see the number 463).
348 364 800
the fourth number which is equal to the product of the factorials of its digits
in base 8: 348 364 000 = [2, 4, 6, 0, 7, 2, 0, 0, 0, 0]8 (see the number 17 280).
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