Those Fascinating Numbers 101

690

• the fourth number n such that σ3(n) is a perfect square: indeed, σ3(690) =

196562 (see the number 345).

693

• the second number n such that Eσ(n) := σ(n + 1) − σ(n) satisfies Eσ(n + 1) =

Eσ(n): here the common value of Eσ is −204, since σ(693) = 1248, σ(694) =

1044 and σ(695) = 840; the sequence of numbers satisfying this property begins

as follows: 44, 693, 3 768 373, 6 303 734, 15 913 724, 20 291 270, . . .

697 (= 17 · 41)

• the smallest 12-hyperperfect number: a number n is said to be 12-hyperperfect

if n = 1 + 12

d|n

1dn

d, which is equivalent to 12σ(n) = 13n + 11: the smallest

12-hyperperfect numbers are 697, 2 041, 1 570 153, 62 722 153, 10 604 156 641

and 13 544 168 521; the number 1 792 155 938 521 is also 12-hyperperfect (see

the number 21).

700

• the smallest multiple of 100 which initiates a sequence of 100 consecutive num-

bers not containing any twin prime pair.

701

• the largest solution x of the diophantine equation x2 + 119 = 15 · 2n: the

solutions (x, n) of this diophantine equation are (1,3), (11,4), (19,5), (29,6),

(61,8) and (701,15): see J. Stiller [190].

707

• the 12th number k such that k|(10k+1 − 1) (see the number 303).

713

• the number of 3-powerful numbers ≤

107;

if N3(x) stands for the number of

3-powerful numbers ≤ x, then we have the

following106

table:

106Here

is how one can create such a table. First of all, observe that each 3-powerful number n

can be written in a unique way as n = s3r4m5, where µ2(r) = µ2(m) = 1 and (r, m) = 1. This is

why

N3(x) =

s3r4m5≤x

(r,m)=1

µ2(r)µ2(m)

=

m5≤x

µ2(m)

r4≤x/m5

(r,m)=1

µ2(r)

s3≤x/(r4m5)

1

=

m≤x1/5

µ2(m)

r≤(x/m5)1/4

(r,m)=1

µ2(r)

x

r4m5

1/3

.