Those Fascinating Numbers 229
81 619
the only five digit number (= 10 000, 20 000, 30 000) whose square contains only
two distinct digits: 81 6192 = 6 661 661 161 (see the number 109).
81 770
the smallest number n which can be written as the sum of the squares of two
prime numbers in 5 (as well as 6, 7 and 8) distinct ways: 81 770 =
412 +2832
=
532
+
2812
=
712
+
2772
=
972
+
2692
=
1372
+
2512
=
1572
+
2392
=
1792
+
2232
=
1932
+
2112
(see the number 338).
82 134
the eighth number n 2 such that
σ(n) + φ(n)
γ(n)2
is an integer (see the number
588).
83 160
the
29th
highly composite number (see the number 180).
83 521 (=
174)
possibly the only fourth power b such that a+b = c, with (a, b) = 1, min(λ(a), λ(c))
3: here a = 857 375, b = 83 521, c = 940 896 and
857 375 + 83 521 = 940 896
53
·
193
+
174
=
25
·
35
·
112
with λ(a) = 3, λ(b) = 4 and λ(c) 3.283.
84 998
the smallest number n which allows the sum
m≤n
1
φ(m)
to exceed 22.
85 139
the third number n such that φ(n) = 4φ(n + 1) (see the number 629).
85 632
the fifth number n such that φ(n) + σ(n) = 3n (see the number 312).
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