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) =

∑

pα 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).