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).