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.