24 Jean-Marie De Koninck
72
the smallest number m such that equation σ(x) = m has exactly five solutions,
namely 30, 46, 51, 55 and 71;
the only solution n
1012
of σ(n) = 2n + 51.
73
the smallest number r which has the property that each number can be written
in the form x1 6 + x2 6 + . . . + xr, 6 where the xi’s are non negative integers (see the
number 4);
the smallest number n such that π(n)
n
log n
+
n
log2
n
, namely the first two
terms of the asymptotic expansion of Li(n): here we have π(73) = 21 while
n
log n
+
n
log2 n
n=73
20.9802; if we let nk be the smallest integer n such that
π(n)
k
j=1
(j 1)!n
logj
n
,
then n1 = 7, n2 = 73, n3 = 1 627, n4 = 230 387, n5 = 12 870 973, n6 =
3 736 935 913 and n7 = 330 645 100 273; on the other hand, by examining a
table of prime numbers, one can verify that 1018 n8 1019;
the smallest number 1 which is equal to the sum of the squares of the
factorials of its digits in base 7: here 73 = [1, 3, 3]7 = 1!2 + 3!2 + 3!2; the only
numbers satisfying this property are 1, 73 and 1 051 783 (see also the number
582).
74
the smallest solution of σ(n + 7) = σ(n) + 7; the only other solution n
108
is
531 434.
75
the fourth horse number (see the number 13).
76
the largest known number n such that the two corresponding numbers (n!)2 +
n! + 1 and (n!)2 + 1 are prime: these two prime numbers both have 223 digits;
the other known numbers n satisfying this property are 1, 2, 3 and 4;
the largest two digit number n which is automorphic, that is whose square ends
with n: here 762 = 5776; the automorphic numbers smaller than 107 are 25,
76, 376, 625, 9376, 90625, 109376, 890625, 2890625 and 7109376.
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