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).