220 Jean-Marie De Koninck
49 372
the smallest number n which allows the sum
m≤n
1
σ(m)
to exceed 8 (see the
number 129).
50 069
the third prime number of the form
11
+
22
+ . . . +
nn,
here with n = 6 (see
the number 3 413).
50 207
the sixth term of the sequence of prime numbers (qn)n≥1 built in the following
manner: set q1 = 2, and having determined qn−1, let qn be the largest prime
factor of 1 + q1q2 . . . qn−1; thus the first 12
terms164
of this sequence are 2, 3, 7,
43, 139, 50207, 340999, 2365347734339, 4680225641471129, 1368845206580129,
889340324577880670089824574922371 and
21087429555133630706328252682943205165142397321.
50 400
the 27th highly composite number (see the number 180).
50 521
the tenth Euler number (see the number 272).
50 624
the smallest number n such that n and n + 1 each have eight prime factors
counting their multiplicity: 50 624 = 26 · 7 · 113 and 50 625 = 34 · 54 (see the
number 135).
50 652
the only solution n 2 · 108 of f(n) = f(n + 1) = f(n + 2), where f(n) =
P (n) + P2(n), with P2(n) standing for the second largest prime factor of n:
here 50 652 = 22 · 33 · 7 · 67, 50 653 = 373 and 50 654 = 2 · 19 · 31 · 43, while the
common value of f(n + i) is 74 (compare with the number 89 460 294).
164This
sequence was first considered by A.A. Mullin [145] who was asking, amongst other things,
if this sequence was increasing, which as it turned out was not since q10 q9. It is possible that the
sequence (qn)n≥1 contains all the prime numbers, but no one was ever successful in proving this
(see W. Narkiewicz [148], p. 2).
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