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


log log x
log x
1/4
Q(x), where Q(x) = min
0σ∞


n=1
1

γ(n)
.
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