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).