Those Fascinating Numbers 125

1 324

• the smallest number which is equal to the sum of the fifth powers of its digits

added to the product of its digits: the only numbers128 satisfying this property

are 1 324, 4 150, 16 363, 93 084 and 97 247.

1 331

• the largest known cube which contains no more than two distinct digits other

than the digit 0; the only known cubes satisfying this property are 27, 64, 343

and 1 331 (see also the number 6 661 661 161);

• the smallest number with an index of composition ≥ 3 which can be written as

the sum of two numbers each with an index of composition ≥ 3: here

1 331 =

113

=

34

+ 2 ·

54,

these last three numbers having as index of composition 3, 4 and 3.09691 re-

spectively; the sequence of numbers satisfying this property begins as follows:

1331, 10935, 18225, 50653, 85293, 357911, 658503, 703125, 711828, 720896,

778688, . . .

1 333

• the smallest 18-hyperperfect number: the four smallest 18-hyperperfect num-

bers are 1 333, 1 909, 2 469 601 and 893 748 277; the number 8 992 165 119 733 is

also 18-hyperperfect (see the number 21).

1 334

• the fourth solution of σ(n) = σ(n + 1) (see the number 14).

1 351

• the seventh number n such that n2 − 1 is powerful (see the number 485): here

1 3512 − 1 = 24 · 33 · 52 · 132;

• the 11th number n for which the distance from en to the nearest integer is the

smallest (see the numbers 178 and 58).

128Let n be a number with this property and whose digits are d1, d2, . . . , dr . We must therefore

have

n =

d5

1

+

d5

2

+ . . . + dr

5

+ d1d2 . . . dr .

Since for each n = [d1, d2, . . . , dr ] 9, we must have n − d1d2 . . . dr ≥

10r−1,

it follows that

10r−1

≤ n − d1d2 . . . dr =

d5

1

+

d5

2

+ . . . + dr

5

≤ r ·

95,

which does not hold if r ≥ 7. It follows that n cannot have more than six digits and must therefore

be smaller than 106. Using a computer, one easily finds the five numbers mentioned above.