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).