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.
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