Those Fascinating Numbers 199
19 601
the ninth number n such that n2 1 is powerful: here 19 6012 1 = 25 · 34 · 52 ·
72 · 112 (see the number 485).
19 661
the smallest prime number which is preceded by 51 consecutive composite num-
bers; indeed, there are no primes between 19 609 and 19 661.
19 683 (=273)
the fifth (and largest) number n whose sum of digits is equal to
3

n (see the
number 512).
19 767 (= 3 · 11 · 599)
the number n which allows the sum
m≤n
Ω(m)=3
1
m
to exceed 2 (see the number 402).
19 841
the smallest prime factor of
1032
+ 1; if qk stands for the smallest prime factor
of
102k
+ 1, then q1 = 101, q2 = 73, q3 = 17, q4 = 353, q5 = 19 841,
q6 = 1 265 011 073, q7 = 257, q8 = 10 753, q9 = 1 514 497,
q10 = 1 856 104 284 667 693 057, q11 = 106 907 803 649 and q12 = 458 924 033;
here are the complete
factorizations157
of the numbers
102k
+1 when 2 k 8
and their partial factorizations when k = 9, 10, 11, 12:
104
+ 1 = 73 · 137,
108
+ 1 = 17 · 5882353,
1016
+ 1 = 353 · 449 · 641 · 1409 · 69857,
1032
+ 1 = 19841 · 976193 · 6187457 · 834427406578561,
1064
+ 1 = 1265011073 · 15343168188889137818369
·515217525265213267447869906815873,
10128
+ 1 = 257 · 15361 · 453377 · P116,
10256
+ 1 = 10753 · 8253953 · 9524994049 · 73171503617 · P225,
10512
+ 1 = 1514497 · 302078977 · C498,
101024
+ 1 = 1856104284667693057 · C1006,
102048
+ 1 = 106907803649 · C2037,
104096
+ 1 = 458924033 · C4088,
157It
is easy to establish that each prime factor of
102k
+ 1 is of the form j ·
2k+1
+ 1 for a certain
positive integer j.
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