Those Fascinating Numbers 335

779 888 018

• the smallest number which can be written as the sum of three distinct fourth

powers in seven distinct ways:

779 888 018 =

34

+

1394

+

1424

=

94

+

384

+

1674

=

144

+

1334

+

1474

=

434

+

1144

+

1574

=

474

+

1114

+

1584

=

634

+

984

+

1614

=

734

+

894

+

1624

(see the number 6 578).

803 685 120

• the ninth number n such that φ(n) + σ(n) = 4n (see the number 23 760).

807 905 281

• the smallest prime number q such that Ω(q + 1) = 2, Ω(q + 2) = 3, . . . ,

Ω(q + 6) = 7 (see the number 61).

817 831 056

• the second number n such that max(β(n), β(n +1), β(n +2)) − min(β(n), β(n +

1), β(n + 2)) = 1; indeed,

817 831 056 =

24

· 3 · 7 · 23 · 97 · 1091 and β(817831056) = 1223,

817 831 057 = 31 · 89 · 461 · 643 and β(817831057) = 1224,

817 831 058 = 2 · 11 ·

1912

· 1019 and β(817831058) = 1223;

the smallest number satisfying this property is 152; the numbers 5 270 522 504,

13 169 880 703, 59 769 918 258, 103 663 433 874 and 475 534 465 837 also satisfy

this property; see the number 89 460 294 for three consecutive numbers with

the same sum of prime factors.

825 753 601

• the smallest prime factor of F16 =

2216

+ 1, whose partial factorization200 is

given by

F16 = 825753601 · 188981757975021318420037633 · C19694.

200Eisenstein had conjectured that all numbers of the form

222··2

+1 were prime. In 1953, Selfridge

proved that this conjecture was false by establishing that F16 (a 19729 digit number) was composite,

its smallest prime factor being 825 753 601.