Those Fascinating Numbers 259
414 645
the fifth solution of equation σ(n) = σ(n + 69) (see the number 8 786).
416 640
the smallest number n such that σ(n) = 17 · φ(n): here 416 640 = 27 · 3 · 5 · 7 · 31,
σ(416640) = 1566720 and φ(416640) = 92160; if nk stands for the smallest
number n such that σ(n) = k · φ(n), then n2 = 3, n3 = 2, n4 = 14, n5 = 56,
n6 = 6, n7 = 12, n8 = 42, n9 = 30, n10 = 168, n11 = 2 580, n12 = 210,
n13 = 630, n14 = 420, n15 = 840, n16 = 20 790, n17 = 416 640, n18 = 9 240,
n19 = 291 060, n20 = 83 160, n21 = 120 120 and n22 = 5 165 160.
417 162
the only number n
109
such that B(n) = B(n + 1) = B(n + 2), where
B(n) =

n
αp: here 417 162 = 2 · 3 · 251 · 277, 417 163 = 17 · 53 · 463,
417 164 =
22
· 11 · 19 · 499, and the common value of B(n + i) is 533 (see the
number 89 460 294).
419 904 (=
26
·
38)
the second number n having at least two distinct prime factors and such that
β(n)3|B1(n):
here (2 +
3)3|(26
+
38)
(see the number 5 120).
421 590 (= 2 · 3 · 5 · 13 · 23 · 47)
the ninth ideal number (see the number 390).
422 481
the smallest number whose fourth power can be written as the sum of three
non zero fourth powers:
422
4814
= 95
8004
+ 217
5194
+ 414
5604;
this is an observation made by R. Frye around 1988; Euler believed that there
were no such
numbers181.
430 272
the third solution of σ(n) = 3n + 12 (see the number 780).
181Euler wrote: Just as there are no non trivial solutions of equation x3 + y3 = z3, there are none
for equations
x4
+
y4
+
z4
=
u4
and
x5
+
y5
+
z5
+
u5
=
v5,
and so on for higher powers. For the
fifth powers, a counter example was found (see the number 144).
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