212 Jean-Marie De Koninck
35 640
the smallest solution of
σ(n)
n
=
11
3
: the only solutions n
109
of this equation
are 35 640, 199 584, 2 142 720 and 12 999 168.
35 677
the smallest prime number q such that

p≤q
p is a multiple of 1 000: here

677
p = 63 731 000; if qk stands for the smallest prime number q such that
∑p≤35
p≤q
p is a multiple of
10k,
then q1 = 5, q2 = 23, q3 = 35 677, q4 = 106 853,
q5 = 632 501 and q6 = 31 190 879.
35 853
the smallest five digit palindrome abcba whose square contains all ten digits
(W. Rutherford, 1835); the numbers 84 648 and 97 779 also have this property
(see the number 32 043).
36 100
the fifth number n such that σ(n) and σ2(n) have the same prime factors,
namely the primes 3, 7, 31 and 127 (see the number 180).
36 551
the first term of the smallest sequence of eight consecutive prime numbers all
of the form 4n + 3 (see the number 463).
36 779
the
39th
Lucas prime number (see the number 613).
37 026 (= 2 ·
32
·
112
· 17)
the smallest number having more than three distinct prime factors and which
is divisible by the square of the sum of its prime factors: the sequence of num-
bers satisfying this property begins as follows: 37 026, 74 052, 81 900, 96 250,
111 078, 121 380, 123 930, . . . ; if nk stands for the smallest number with more
than three prime factors and which is divisible by the k-th power of the sum
of its prime factors, then n1 = 1 122, n2 = 37 026 and n3 = 1 221 858.
37 717
the prime number which appears the most often as the
19th
prime factor of an
integer (see the number 199).
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