Those Fascinating Numbers 51
173
the second three digit number = 100, 200, 300, and whose square has only two
distinct digits: 1732 = 29 929 (see the number 109).
174
the seventh number k for which equation a1a2 . . . ak = a1 + a2 + . . . + ak
has exactly one solution (namely a1 = 2, a2 = k, ai = 1 for 3 i k):
M. Misiurewicz [138] proved that the only numbers k 50 000 satisfying this
property are 2, 3, 4, 6, 24, 114, 174 and 444 (see also R.K. Guy [101], D24).
175
the third number n 9 such that n =
∑r
i=1
di, i where d1, . . . , dr stand for the
digits of n: here 175 = 11 + 72 + 53; there seems to exist only nine numbers
satisfying this property, namely 89, 135, 175, 518, 598, 1 306, 1 676, 2 427 and
2 646 798;
the value of the sum of the elements of a diagonal, of a line or of a column in
a 7 × 7 magic square (see the number 15).
177
the smallest number n which allows the sum
m≤n
1
φ(m)
to exceed 10; if we
denote by nk the smallest number n that allows this sum to exceed k, then it
follows from the asymptotic formula (due to H.L. Montgomery [142])
m≤n
1
φ(m)
= c log n d + O
log n
n
,
where
c =
p
1 +
1
p(p 1)
=
ζ(2)ζ(3)
ζ(6)
1.9436
stands for the Riemann Zeta Function) and
d = γ

n=1
µ2(n)
nφ(n)


n=1
µ2(n)
log n
nφ(n)
0.0605,
that nk nk :=
[e(k+d)/c];
the following table compares the effective value nk
with the predicted value nk (which as a matter of fact turns out to be fairly
accurate):
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