158 Jean-Marie De Koninck
3 767
the smallest number n such that

m≤n
σ(m) is a multiple of 10 000 (here the
sum is equal to 11 670 000).
3 775
the sixth number n such that 2n 7 is prime (see the number 39).
3 777
the smallest number n such that σ(n), σ(n + 1), σ(n + 2) and σ(n + 3) have
the same prime factors, namely here 2, 3, 5 and 7: the sequence of numbers
satisfying this property begins as follows: 3 777, 6 044, 20 154, 20 155, 29 395,
29 396, . . . ; if nk stands for the smallest number n such that σ(n), σ(n + 1),
. . . , σ(n+k −1) have the same prime factors, then n2 = 5, n3 = 33, n4 = 3 777,
n5 = 20 154 et n6 = 13 141 793.
3 823
the smallest prime number q such that

p≤q
p is divisible by 210 (= 2 · 3 · 5 · 7):
here this sum is equal to 939 330 (see the number 269).
3 879
the smallest number n which allows the sum
m≤n
1
φ(m)
to exceed 16 (see the
number 177).
3 901
the smallest 30-hyperperfect number: 3 901 and 28 600 321 are the only two
30-hyperperfect numbers 2 ·
109
(see the number 21).
3 906
the smallest number n such that
n6
+ 1 091 is a prime number, a result with a
historical importance because it was obtained by Daniel Shanks before the 60’s,
namely before the computer age (see H.C. Williams [205]); the only numbers
n 10 000 whose corresponding number
n6
+ 1 091 is
prime147
are 3 906, 4 620,
5 376, 5 460 and 8 190.
147One easily sees that in order for n6 + 1 091 to be prime, n must be a multiple of 2, 3 and 7.
Previous Page Next Page