136 Jean-Marie De Koninck

1 782 (= 2 · 34 · 11)

• the smallest number n 1 such that σ(n) = γ(n)2, and the only one smaller

than 109: it is easy to see that such a number n 1 must be even134 and cannot

be square-free (compare with the number 96, as well as with the number 108);

• the fourth positive solution x of the diophantine equation

x2

+ 999 =

y3

(see

the number 251).

1 792

• the

11th

Granville number (see the number 126).

1 798

• the fifth solution of σ(φ(n)) = σ(n) (see the number 87).

1 807

• the fifth voracious number: consider the sequence (bk)k≥1 defined by

b1 = 2, bk+1 = 1 + b1b2 . . . bk = bk

2

− bk + 1 (k = 1, 2, . . .);

each term of this sequence is called a voracious number; thus the first voracious

numbers are 2, 3, 7, 43, 1 807, 3 263 443, 10 650 056 950 807,

113 423 713 055 421 844 361 000 443,

12 864 938 683 278 671 740 537 145 998 360 961 546 653 259 485 195 807, . . . ; it is

interesting to mention that

∑∞

j=1

1/bj = 1; for more on this subject, see the

recent papers of G. Myerson & J.W. Sander [146] and of J.W. Sander [179].

1 847

• the smallest number n such that

∑

m≤n

σ(m) is a multiple of 1 000: here the

sum is equal to 2 805 000.

1 848

• the largest convenient number (see the number 37); the only known convenient

numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30,

33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130,

133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385,

408, 462, 520, 760, 840, 1320, 1365 and 1848; S. Chowla [33] proved that there

is only a finite number of convenient numbers135.

134Indeed, if n 1 is odd, then γ(n)2 = σ(n) is also odd, so that n = m2 for a certain m, in which

case n σ(n) =

γ(n)2

=

γ(m2)2

=

γ(m)2

≤

m2,

a contradiction.

135On the other hand, it is known that if one could find another one, there would be no others.