Those Fascinating Numbers 87

439

• the smallest prime factor of the Mersenne number 273 − 1, whose complete

factorization is given by

273

− 1 = 439 · 2298041 · 9361973132609.

441

• the smallest solution of τ (n + 9) = τ (n) + 9; the sequence of numbers satisfying

this equation begins as follows: 441, 775, 841, 1681, 2907, 3969, 4087, . . .

444

• the largest number k 50 000 for which equation a1a2 . . . ak = a1 +a2 +. . .+ak

has exactly one solution (see the number 174).

454

• the largest number which cannot be represented as the sum of less than eight

cubes of non negative integers90.

455 (= 5 · 7 · 13)

• the

100th

number having exactly three distinct prime factors; if

pnk) (

stands for

the

nth

number having exactly k distinct prime factors, we have the following

table:

n pn

(1)

pn

(2)

pn

(3)

pn

(4)

pn

(5) pn(6)

10 16 24 105 660 6006 67830

102

419 184 455 2262 17490 184470

103

7517 2102 2988 10836 67860 617610

104

103511 26608 26128 67512 331674 2537964

105

1295953 322033 258764 510873 1980902 12580890

106

15474787 3741154 2677258 4357756 13757850 72789420

107

179390821 42314023 28013887 39780102 106254070 476495370

459

• the smallest number n such that f(n) = f(n + 1), where f(n) =

β(n)

ω(n)

: here

459 =

33

· 17, 460 =

22

· 5 · 23 and f(459) = f(460) = 10; the sequence of num-

bers satisfying this property begins as follows: 459, 2023, 5063, 11111, 87615,

92080, 224720, 268191, 390224, 524799, 535601, 680096, 758848,. . . (compare

with the number 735).

90Using a computer, one easily checks that the only numbers 1000 which cannot be written as

the sum of seven cubes (or less) of non negative integers are 15, 22, 23, 50, 114, 167, 175, 186, 212,

231, 238, 239, 303, 364, 420, 428 and 454.