104 Jean-Marie De Koninck
735
the smallest number n such that f(n) = f(n + 1), where f(n) =
B(n)
Ω(n)
: here
735 = 3 · 5 ·
72,
736 =
25
· 23 and f(735) = f(736) = 11/2; the sequence
of numbers satisfying this property begins as follows: 735, 2172, 3484, 6324,
6747, 6867, 7424, 7865, 14012, 14640, 35321, 39284, 42172, 47724, 57075, 60155,
63664, 89975, . . . (compare with the number 459);
the only odd number n
1011
having at least two digits, having all its digits
different from 1 and 0, and whose sum of digits as well as the product of its
digits divides n (see the numbers 24 and 111).
736
the only number of the form abc such that abc = a +
bc;
indeed, here we have
736 = 7 +
36.
740
the second number n such that ω(n) = ω(n + 1) = ω(n + 2) = 3: here 740 =
22 · 5 · 37, 741 = 3 · 13 · 19, 742 = 2 · 7 · 53 (see the number 644).
742
the eighth Keith number (see the number 197).
744
the sixth number n such that σ(φ(n)) = n; the sequence of numbers satisfying
this equation begins as follows: 1, 3, 15, 28, 255, 744, 2418, 20 440, 65 535,
548 856, . . . (in particular, if n = Fk 2 where Fk =
22k
+ 1 is a Fermat prime,
then n is one of these numbers).
748
the second number n such that σ(n) = 2n + 16 (see the number 550).
750
the number of pseudoprimes in base 2 smaller than 107 (see the number 245).
751
the 14th number n such that n · 2n 1 is prime (see the number 115).
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