210 Jean-Marie De Koninck

32 043

• the smallest number whose square uses each of the ten digits once and only

once: 87 numbers satisfy this property, namely: 32043, 32286, 33144, 35172,

35337, 35757, 35853, 37176, 37905, 38772, 39147, 39336, 40545, 42744, 43902,

44016, 45567, 45624, 46587, 48852, 49314, 49353, 50706, 53976, 54918, 55446,

55524, 55581, 55626, 56532, 57321, 58413, 58455, 58554, 59403, 60984, 61575,

61866, 62679, 62961, 63051, 63129, 65634, 65637, 66105, 66276, 67677, 68763,

68781, 69513, 71433, 72621, 75759, 76047, 76182, 77346, 78072, 78453, 80361,

80445, 81222, 81945, 83919, 84648, 85353, 85743, 85803, 86073, 87639, 88623,

89079, 89145, 89355, 89523, 90144, 90153, 90198, 91248, 91605, 92214, 94695,

95154, 96702, 97779, 98055, 98802 and

99066.161

32 045

• the second number which can be written as the sum of two squares in eight

distinct ways: 32 045 =

22

+

1792

=

192

+

1782

=

462

+

1732

=

672

+

1662

=

742

+

1632

=

862

+

1572

=

1092

+

1422

=

1222

+

1312

(see the number 50).

32 214

• the third number n such that each of the numbers n + i, i = 0, 1, 2, . . . , 16, has

a factor in common with the product of the other 16 (see the number 2 184).

32 292

• the sixth number n 1 such that n · 2n + 1 is prime (see the number 141).

32 445

• the second number n such that σ(n) = 2n + 6 (see the number 8 925).

32 469

• the seventh number n 1 such that n ·

2n

+ 1 is prime (see the number 141).

32 760

• the second number n such that σ(n) = 4n (see the number 30 240).

32 768

• the eighth number n 1 such that φ(σ(n)) = n (see the number 128).

161Observe that none of these numbers is prime; indeed, each of them is a multiple of 3: this results

from the fact that to each such number n corresponds the number

n2

which is made up of the digits

0,1,2,. . . ,9; since 0 + 1 + 2 + . . . + 9 = 45 is divisible by 9, it follows that 3|n.