Those Fascinating Numbers 249

217 854

• the third number n such that (σI (n) + γ(n))/n is an integer (see the number

270).

218 295

• the smallest number n which allows the sum

m≤n

1

σ(m)

to exceed 9 (see the

number 129).

218 984

• the fourth composite number n such that σ(n + 8) = σ(n) + 8 (see the number

1 615).

219 978

• the sixth number which is not a palindrome, but which divides the number

obtained by reversing its digits (see the number 1 089).

223 232

• the number n which allows176 the sum G(n) :=

m≤n

1

γ(m)

to exceed 100: here

G(223232) ≈ 100.000549.

225 504

• the second number n such that β(n)|β(n + 1) and β(n + 1)|β(n + 2), where

β(n) =

∑

p|n

p: here we have 34|408 and 408|2448; the sequence of num-

bers satisfying this property begins as follows: 24, 225 504, 944 108, 10 869 375,

11 506 989, 12 792 675, 20 962 395, 25 457 760, 79 509 528, 89 002 914, 89 460 294,

146 767 704, . . .

228 479

• the smallest prime factor of the Mersenne number

271

− 1, whose complete

factorization is given by

271

− 1 = 228 479 · 48 544 121 · 212 885 833.

176In

1962, N.G. de Bruijn [43] proved that log G(x) ∼ 2 2 log x/ log log x as x → ∞; in 1965,

W. Schwarz [183] obtained an asymptotic formula for G(x), namely

G(x) = (1 + o(1))

1

21/4

√

4π

log log x

log x

1/4

Q(x), where Q(x) = min

0σ∞

xσ

∞

n=1

1

nσ

γ(n)

.