Those Fascinating Numbers 167

5 395

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

5 407

• the sixth prime number q such that

∑

p≤q

p is a multiple of 100: here this sum

is equal to 1 776 800 (see the number 563).

5 472

• the second dihedral 3-perfect number n, that is such that τ (n)+ σ(n) = 3n: the

only three numbers n 109 satisfying this property are 60, 5 472 and 2 500 704.

5 525 (=

52

· 13 · 17)

• the smallest number which can be written as the sum of two squares in five

distinct ways (as well as six distinct ways), namely 5 525 =

72

+

742

=

142

+

732

=

222

+

712

=

252

+

702

=

412

+

602

=

502

+

552

(see the number 50).

5 569

• the largest known150 prime number p for which there exists an even number

n (namely here n = 389 965 026 819 938) such that n − q is composite for each

prime number q p (see J. Richstein [171]): here we have the “Goldbach

representation” 389 965 026 819 938 = 5 569 + 389 965 026 814 369.

5 597

• the number of pseudoprimes in base 2 smaller than

109

(see the number 245).

5 694

• the smallest number n such that βI (n) = βI (n+1) = βI (n+2), where βI (n) :=

∑

p|n,p2

p: here 5 694 = 2 · 3 · 13 · 73, 5 695 = 5 · 17 · 67 and 5 696 =

26

· 89, while

the common value of βI (n + i) is 89; the second number with this property is

2 463 803 977 (see also the number 89 460 294).

5 719

• the sixth Lucas-Carmichael number(see the number 399).

150It is possible to prove that there exist infinitely many prime numbers p satisfying this property.