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).
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