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.
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