50 Jean-Marie De Koninck
168
the smallest number m such that equation σ(x) = m has exactly six solutions,
namely 60, 78, 92, 123, 143 and 167;
the number of prime numbers 1000;
the largest known number k such that the decimal expansion of 2k does not
contain the digit 2; indeed, using a computer, one can verify that
2168
= 374144419156711147060143317175368453031918731001856
does not contain the digit 2, while each number
2k,
for k = 169, 170, . . . , 3000,
contains it (see the number 71);
the smallest solution of σ2(n) = σ2(n + 14); the sequence of numbers satisfying
this equation begins as follows: 108, 785, 7553, 6632633, 9535673, . . .
169
the smallest perfect square m3 2 for which there exist numbers m1 and m2 such
that mi 2 (mi 1)2 = mi−1 2 for i = 2, 3: here 169 = 132 = 122 + 52 =
122 + 42 + 32.
170
the smallest number n such that λ0(n) = λ0(n + 1) = . . . = λ0(n + 6) = −1,
where λ0 stands for the Liouville function;
the smallest number n 1 such that φ(n)σ(n)
is63
a fourth power: here
φ(n)σ(n) =
124;
the sequence of numbers satisfying this property begins as
follows: 1, 170, 595, 714, 121056, 480441, 529620, 706063, 706237, 729752,
755972, 815654, . . .
171
the largest known number n such that the binomial coefficient
(2n)
n
is not
divisible by the square of an odd prime number (R.K. Guy [101], B33): here
342
171
=
25
· 3 · 5 · 11 · 17 · 29 · 31 · 37 · 43 · 47 · 59 · 61 · 67 · 89 · 97 · 101
·103 · 107 · 109 · 113 · 173 · 179 · 181 · 191 · 193 · 197 · 199
·211 · 223 · 227 · 229 · 233 · 239 · 241 · 251 · 257 · 263 · 269
·271 · 277 · 281 · 283 · 293 · 307 · 311 · 313 · 317 · 331 · 337.
63One
can prove that there exist infinitely many numbers n such that φ(n)σ(n) is a perfect square.
In fact, Florian Luca can prove (private communication) that, for each positive integer s for which
there exist numbers n0 and x0 such that φ(n0)σ(n0) =
sx2,
0
equation φ(n)σ(n) =
sx2
has infinitely
many solutions (n, x).
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