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

(ζ 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):