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.
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