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).