84 Jean-Marie De Koninck

400

• the smallest multiple of 100 for which the following 100 numbers include exactly

17 prime numbers, namely 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461,

463, 467, 479, 487, 491 and 499; the sequence of multiples of 100 satisfying this

property begins as follows: 400, 1 400, 783 700, 2 704 900, . . .

402 (= 2 · 3 · 67)

• the smallest number n which allows the sum

m≤n

Ω(m)=3

1

m

to exceed 1; if nk stands

for the smallest number n which allows this sum to exceed k, then the sequence

(nk)k≥1 begins as follows: 402, 19 767, 1 023 587, . . .

403

• the smallest number n such that

∑

m≤n

σ(m) is a multiple of 100 (here the sum

is equal to 133 600).

406

• the number of integer zeros of the function M(x) :=

n≤x

µ(n) located in the

interval [1, 10 000] (see the number 92);

• the smallest solution of σ(n) = σ(n + 29); the sequence of numbers satisfying

this equation begins as follows: 406, 496, 825, 934, 1215, 1755, 2265, 2685,

2896, . . .

407

• the largest number which can be written as the sum of the cubes of its digits:

407 = 43 + 03 + 73; the others are 1, 153, 370 and 371;

• the

11th

number k such that

k|(10k+1

− 1) (see the number 303).

408

• the fourth solution y of the Fermat-Pell equation x2 − 2y2 = 1: here (x, y) =

(577, 408) (see the number 99).

409

• the rank of the prime number which appears the most often as the 13th prime

factor of an integer: p409 = 2083 (see the number 199).