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.