Those Fascinating Numbers 289

4 563 000

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

also the number 312).

4 620 799

• the largest known prime number p such that P (p2 −1) ≤ 31 and P (p2 −1) = 31:

here p2 − 1 = 210 · 32 · 52 · 72 · 132 · 192 · 31 (see the number 4 801).

4 681 203 (= 3 · 37 · 181 · 233)

• the smallest square-free composite number n such that p|n =⇒ p + 12|n + 12

(see the number 399).

4 695 456

• the

12th

number n such that φ(n) + σ(n) = 3n (see the number 312).

4 713 984 (=

29

·

33

· 11 · 31)

• the fifth solution of

σ(n)

n

=

10

3

(see the number 1 080).

4 729 494

• the number appearing in the famous “cattle problem” of Archimedes, namely

in the Fermat-Pell equation

x2

− 4 729 494

y2

= 1 (see J. Stillwell [191]).

4 737 595

• the fourth solution of σ2(n) = σ2(n + 10) (see the number 120).

4 989 191

• the smallest number n which allows the sum

i≤n

1

i

to exceed 16 (see the number

83).

5 058 180

• the fourth even number n such that σI (n) = σI (n +2) (see the number 54 178).