Those Fascinating Numbers 185

10 693

• the smallest 11-hyperperfect number (and the only one

108):

a number

n is said to be 11-hyperperfect if n = 1 + 11

d|n

1dn

d, which is equivalent to

11σ(n) = 12n + 10 (see the number 21).

10 790

• the smallest solution of σ2(n) = σ2(n + 20); the sequence of numbers satisfying

this equation begins as follows: 10 790, 9 475 190, 13 622 390, 22 302 710, . . .

10 854

• the smallest number n which allows the sum

m≤n

1

φ(m)

to exceed 18 (see the

number 177).

10 891

• the smallest number n such that

∑

m≤n

φ(m) is a multiple of 10 000 (here the

sum is equal to 36 060 000).

10 935

• the sixth odd number n 1 such that γ(n)|σ(n) (see the number 135).

10 989

• the third number which is not a palindrome and which divides the number

obtained by reversing its digits (see the number 1 089).

11 112

• the second insolite number (see the number 111).

11 167

• the smallest number n which allows the sum

m≤n

1

σ(m)

to exceed 7 (see the

number 129).

11 213

• the exponent of the 23rd Mersenne prime 211 213 − 1 (Gillies, 1963).