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.
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