Those Fascinating Numbers 235

99 826

• the smallest number n such that the Liouville function λ0 takes successively,

starting with n, the values

1, −1, 1, −1, 1, −1, 1, −1, 1, −1, 1, −1, 1, −1, 1, −1

(see the number 6 185).

99 954

• the value of the Kaprekar constant for the five digit numbers (see the number

495).

99 991

• the largest five digit prime number.

100 003

• the smallest six digit prime number.

100 823

• the smallest number n such that the decimal expansion of 2n contains nine

consecutive zeros (see the number 53).

102 359

• the smallest prime number made up of six distinct digits (see the number 1 039).

102 510

• the smallest six digit vampire number (see the number 1 260).

102 564

• the smallest number which quadruples when its last digit is moved in first

position: the smallest six numbers satisfying this property are 102 564, 128 205,

153 846, 179 487, 205 128 and 230 769; one can show that there exist

infinitely171

many such numbers, the seventh being 102 564 102 564.

171To prove this, one only needs to observe that by performing a concatenation of each of these

numbers with itself (as long as it is required), one obtains a number with the same property; thus,

the numbers 102 564 102 564 and 102 564 102 564 102 564 are such numbers.