Those Fascinating Numbers 79

368

• the second solution of σ(n) = 2n + 8; the sequence of numbers satisfying

this equation begins as follows: 56, 368, 836, 11 096, 17 816, 45 356, 77 744,

91 388, 128 768, 254 012, 388 076, 2 087 936, 2 291 936, 13 174 976, 29 465 852,

35 021 696,. . .

83

369

• the tenth number k such that k|(10k+1 − 1) (see the number 303).

370

• one of the five numbers which can be written as the sum of the cubes of its

digits: 370 = 33 + 73 + 03; the others are 1, 153, 371 and 407.

371

• one of the five numbers which can be written as the sum of the cubes of its

digits: 371 =

33

+

73

+

13;

the others are 1, 153, 370 and 407.

373

• the largest three digit number (see the number 264) which can be written in two

distinct ways as the sum of positive powers of its digits: 373 =

31

+

73

+

33

=

34

+

72

+

35.

376

• the smallest three digit automorphic number: 3762 = 141 376 (see the number

76).

377

• the largest known Fibonacci pseudoprime; the only other one known is 323;

• the smallest number n such that the decimal expansion of 2n contains four

consecutive zeros (see the number 53).

83It

is easy to show that any number n =

2α

·p, where α is a positive integer such that p =

2α+1

−9

is prime, is a solution of σ(n) = 2n + 8: this is the case when α = 3, 4, 8, 10, 16, 20, 32 (and for no

other values of α ≤ 100); the solutions corresponding to α = 3, 4, 8, 10 are included in the above

list. It is also possible to identify the solutions n of the form 2α · p · q, with p q primes and α

a positive integer. This is how one obtains the solutions n = 836, 11 096, 17 816, 77 744, 2 291 936,

13 174 976 and 35 021 696 listed here.