Those Fascinating Numbers 155
3 391
the smallest prime factor of the Mersenne number
2113
1, whose complete
factorization is given by
2113
1 = 3391 · 23279 · 65993 · 1868569 · 1066818132868207,
this number being the smallest Mersenne number having exactly five prime
factors (see the number 223);
the smallest prime number having two representations of the form
x3
+
2y3:
here 3 391 =
153
+ 2 ·
23
=
93
+ 2 ·
113
(see the number 41 for important
references); the sequence of numbers satisfying this property begins as follows:
3 391, 8 317, 79 939, 593 209, 942 199, 1 229 257,. . .
3 413
the second prime number of the form
∑n
i=1
ii
(here with n = 5); setting sj =
∑j
i=1
ii
and letting nk stand for the
kth
prime number in the sequence (sj )j≥1,
then n1 = s2 = 11 + 22 = 5, n2 = s5 = 3 413, n3 = s6 = 50 069, n4 = s10 =
10 405 071 317 and n5 = s30; no other prime numbers of this form are known144.
3 435
the smallest number n = [d1, d2, . . . , dr] 1 such that
1≤i≤r
di=0
di
di
= n; the only
other number satisfying this property is n = 438 579 088.
3 445 (= 5 · 13 · 53)
the smallest square-free composite number n such that p|n =⇒ p + 11|n + 11.
3 462
the number of solutions 2 x1 x2 . . . x6 of
6
i=1
1
xi
= 1 (see the number
147, as well as R.K. Guy [101], D11).
144It
is easy to establish that all terms s4k−1 and s4k are even, thus not prime. In 1993,
K. Soundararajan [189] proved that if A = {s1, s2, . . .} and if πA(x) = #{p x : p A}, that
is the cardinality of the set of prime numbers p x which belong to A, then πA(x)
log x
(log log x)2
.
This same author conjectures that there exists a constant c 0 such that πA(x) c
log x
(log log x)2
,
and therefore in particular that there exist infinitely many prime numbers of the form
11
+
22
+
33 + . . . + nn.
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