114 Jean-Marie De Koninck

954

• the only three digit self replicating number: a number n all of whose digits are

distinct and decreasing is said to be self replicating if the process of reversing

its digits and subtracting this new number from the number n yields a number

whose digits are the same as that of n: no numbers with 1, 2, 5, 6 or 7 digits

satisfy this property; see M. Gardner [87]; the numbers 954, 7 641, 98 754 210,

987 654 321 and 9 876 543 210 are the only ones satisfying this property.

957

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

959

• the

13th

number k such that

k|(10k+1

− 1) (see the number 303).

960

• the fourth number n divisible by a square 1 and such that γ(n+1)−γ(n) = 1

(see the number 48);

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

cubes of its digits: the only numbers satisfying this property are 12, 30, 666,

870, 960 and 1 998.

968

• the only solution n

1012

of σ(n) = 2n + 59 (see the number 196);

• the second powerful number which can be written as the sum of two co-prime

3-powerful numbers = 1: 968 = 343 + 625, that is

23

·

112

=

73

+

54

(see the

number 841).

971

• the largest irregular prime smaller than 1000 (see the number 59).

974

• the

17th

number n such that n! − 1 is prime (see the number 166).

975

• the tenth solution of φ(n) = φ(n + 1) (see the number 15).