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 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
.
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