358 Jean-Marie De Koninck
93 112 129 088 (= 26 · 112 · 17 · 294)
the number, amongst all those 1011, whose index of composition is the near-
est205 to the number e: here λ(93112129088) 2.7182833 (see the number
7 826 354 460 for the analogue problem with the number π).
99 999 999 977
the largest 11 digit prime number.
100 000 000 003
the smallest 12 digit prime number.
102 564 102 564
the seventh number which quadruples when its last digit is moved in first po-
sition (see the number 102 564).
104 287 176 419 (= 1201 · 86833619)
the 100 000 000
000th
composite number (see the number 133).
111 122 111 232
the
22nd
insolite number (see the number 111).
111 132 122 112
the
23rd
insolite number (see the number 111).
111 211 322 112
the
24th
insolite number (see the number 111).
111 312 122 112
the
25th
insolite number (see the number 111).
205To obtain this result, we proceed as follows. Our focus is the set {n x : e ε λ(n) e + ε}
for a given x (here x =
1011)
and a certain precision ε 0 (say ε =
10−6).
It is clear that this set
is equal to A(x, e ε) \ A(x, e + ε), where A(x, z) = {n x : λ(n) z}. But clearly each element
n A(x, z) is of the form n = ms, where m is powerful, s is square-free, (m, s) = 1 and λ(ms) z,
where this last condition is equivalent to s (m/γ(m)z)1/(z−1).
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