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.
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